INTERNATIONAL GONGRESS 11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z">k. The main point is that the notion of a finite correspondence for smooth finite type schemes over a field extends to a corresponding notion over a general base-scheme (see [ 33 , $ 8 ] [ 33 , $ 8 ] [33,$8][33, \$ 8][33,$8] ). This gives rise to a theory of motivic cohomology generalizing Voevodsky's definition as
H p , q ( X , Z ) := Hom D M C D ( S ) ( M ( X ) , Z S ( q ) [ p ] ) H p , q ( X , Z ) := Hom D M C D ( S ) ⁡ M ( X ) , Z S ( q ) [ p ] H^(p,q)(X,Z):=Hom_(DM_(CD)(S))(M(X),Z_(S)(q)[p])H^{p, q}(X, \mathbb{Z}):=\operatorname{Hom}_{\mathrm{DM}_{\mathrm{CD}}(S)}\left(M(X), \mathbb{Z}_{S}(q)[p]\right)Hp,q(X,Z):=HomDMCD(S)⁡(M(X),ZS(q)[p])
for X X XXX smooth over S S SSS. They show that the assignment S DM C D ( S ) S ↦ DM C D ⁡ ( S ) S|->DM_(CD)(S)S \mapsto \operatorname{DM}_{\mathrm{CD}}(S)S↦DMCD⁡(S) defines a functor to the category of triangulated tensor categories, D M C D ( ) : S c h B o p T r D M C D ( − ) : S c h B o p → T r ⊗ DM_(CD)(-):Sch_(B)^(op)rarrTr^(ox)\mathrm{DM}_{\mathrm{CD}}(-): \mathrm{Sch}_{B}^{\mathrm{op}} \rightarrow \mathbf{T r}^{\otimes}DMCD(−):SchBop→Tr⊗, admitting a sixfunctor formalism. There are also Tate twists M M ( n ) M ↦ M ( n ) M|->M(n)M \mapsto M(n)M↦M(n). This gives a definition of motivic cohomology of an general scheme Y Y YYY by
which for Y S m S Y ∈ S m S Y inSm_(S)Y \in \mathrm{Sm}_{S}Y∈SmS agrees with the definition given above.
They construct an adjunction
ϕ : S H ( Y ) DM C D ( Y ) : ϕ Ï• ∗ : S H ( Y ) ⇄ DM C D ⁡ ( Y ) : Ï• ∗ phi^(**):SH(Y)⇄DM_(CD)(Y):phi_(**)\phi^{*}: \mathrm{SH}(Y) \rightleftarrows \operatorname{DM}_{\mathrm{CD}}(Y): \phi_{*}ϕ∗:SH(Y)⇄DMCD⁡(Y):ϕ∗
with ϕ Ï• ∗ phi_(**)\phi_{*}ϕ∗ playing the role of the Eilenberg-MacLane functor, giving rise to the spectrum M Y S H ( Y ) M Y ∈ S H ( Y ) M_(Y)inSH(Y)\mathcal{M}_{Y} \in \mathrm{SH}(Y)MY∈SH(Y) representing H , ( Y , Z ) H ∗ , ∗ ( Y , Z ) H^(**,**)(Y,Z)H^{*, *}(Y, \mathbb{Z})H∗,∗(Y,Z) [33, DEfInITION 11.2.17]. They discuss the question of whether Y M Z Y Y ↦ M Z Y Y|->MZ_(Y)Y \mapsto \mathcal{M} \mathbb{Z}_{Y}Y↦MZY is cartesian (see [33, CONJECTURE 11.2.22, PROPOSITION 11.4.7]), without reaching a general resolution.
Cisinski-Déglise have a different approach for representing motivic cohomology with Q Q Q\mathbb{Q}Q-coefficients, much in the same spirit as Beilinson's construction of universal cohomology using algebraic K K KKK-theory. Using the spectrum KGL S S H ( S ) KGL S ∈ S H ( S ) KGL_(S)inSH(S)\operatorname{KGL}_{S} \in \mathrm{SH}(S)KGLS∈SH(S), which represents homotopy invariant algebraic K K KKK-theory, they use the Adams operations to decompose K G L S Q K G L S Q KGL_(SQ)\mathrm{KGL}_{S \mathbb{Q}}KGLSQ into summands, K G L S Q = i K G L S ( i ) K G L S Q = ⨁ i   K G L S ( i ) KGL_(SQ)=bigoplus_(i)KGL_(S)^((i))\mathrm{KGL}_{S \mathbb{Q}}=\bigoplus_{i} \mathrm{KGL}_{S}^{(i)}KGLSQ=⨁iKGLS(i), with K G L S ( i ) K G L S ( i ) KGL_(S)^((i))\mathrm{KGL}_{S}^{(i)}KGLS(i) representing the i i iii th graded piece of K K KKK-theory for the γ γ gamma\gammaγ-filtration. This gives them a nice commutative monoid object (i.e., commutative ring spectrum) H S E := K G L S ( 0 ) S H ( S ) Q H S E := K G L S ( 0 ) ∈ S H ( S ) Q H_(S)^(E):=KGL_(S)^((0))inSH(S)_(Q)H_{S}^{\mathrm{E}}:=\mathrm{KGL}_{S}^{(0)} \in \mathrm{SH}(S)_{\mathbb{Q}}HSE:=KGLS(0)∈SH(S)Q, whose module category they call the category of Beilinson motives over S S SSS. This construction is cartesian, gives a good theory of motivic cohomology with Q Q Q\mathbb{Q}Q-coefficients over a general base-scheme and agrees with D M C D ( S ) Q D M C D ( S ) Q DM_(CD)(S)_(Q)\mathrm{DM}_{\mathrm{CD}}(S)_{\mathbb{Q}}DMCD(S)Q for S S SSS a uni-branch scheme. See [33, §14] for details.

3.2. Spitzweck's motivic cohomology

In [110], Spitzweck constructs a motivic cohomology theory over an arbitrary basescheme. The Bloch cycle complex gives rise to a general version of Bloch's higher Chow groups for finite type schemes over a Dedekind domain, which has nice localization properties (by [25] and [84]), but has poor functoriality and lacks a multiplicative structure. On the other hand, using the Bloch-Kato conjectures, established by Voevodsky et al., the ℓ ℓ\ellℓ-completed higher Chow groups are recognized as a truncated ℓ ℓ\ellℓ-adic étale cohomology, for ℓ ℓ\ellℓ prime to all residue characteristics. The theorem of Geisser-Levine [52] describes the p p ppp-completed higher Chow groups in characteristic p > 0 p > 0 p > 0p>0p>0 in terms of logarithmic de RhamWitt sheaves. Finally, there is the good theory with Q Q Q\mathbb{Q}Q-coefficients given by Beilinson motivic cohomology of Cisinski-Déglise, as described above.
Each of these three theories, namely the ℓ ℓ\ellℓ-adic étale cohomology, the cohomology of the logarithmic de Rham-Witt sheaves, and the rational Beilinson motivic cohomology, has good functoriality and multiplicative properties. Gluing the ℓ ℓ\ellℓ-adic, p p ppp-adic, and rational theories together via their respective comparisons with the Bloch cycle complex, Spitzweck constructs a theory with good functoriality and multiplicative properties, and which is described by a presheaf of complexes on smooth schemes over a given Dedekind domain as base-scheme. The corresponding theory agrees with Voevodsky's motivic cohomology for smooth schemes over a perfect field, and is given additively by the hypercohomology of the Bloch complex for smooth schemes over a Dedekind domain (even in mixed characteristic).
Taking the base-scheme to be Spec Z Z Z\mathbb{Z}Z, Spitzweck's construction yields a representing object M Z Z M Z Z MZ_(Z)M \mathbb{Z}_{\mathbb{Z}}MZZ in S H ( Z ) S H ( Z ) SH(Z)\mathrm{SH}(\mathbb{Z})SH(Z) and one can thus define absolute motivic cohomology for smooth schemes over a given base-scheme S S SSS by pulling back M Z Z M Z Z MZ_(Z)M \mathbb{Z}_{\mathbb{Z}}MZZ to M Z S S H ( S ) M Z S ∈ S H ( S ) MZ_(S)inSH(S)M \mathbb{Z}_{S} \in \mathrm{SH}(S)MZS∈SH(S). The resulting motivic cohomology agrees with Voevodsky's for smooth schemes of finite type over a perfect base-field, and with the hypercohomology of the Bloch cycle complex for smooth finite type schemes over a Dedekind domain. This gives rise to a triangulated category of motives D M s p ( S ) D M s p ( S ) DM_(sp)(S)\mathrm{DM}_{\mathrm{sp}}(S)DMsp(S) over a base-scheme S S SSS, defined as the homotopy category of M Z S M Z S − MZ_(S^(-))M \mathbb{Z}_{S^{-}}MZS− modules, and the functor S D M S p ( S ) S ↦ D M S p ( S ) S|->DM_(Sp)(S)S \mapsto \mathrm{DM}_{\mathrm{Sp}}(S)S↦DMSp(S) inherits a Grothendieck six-functor formalism from that of S S H ( S ) S ↦ S H ( S ) S|->SH(S)S \mapsto \mathrm{SH}(S)S↦SH(S).

3.3. Hoyois' motivic cohomology

Spitzweck's construction gives a solution to the problem of constructing a triangulated category of motives over an arbitrary base, admitting a six-functor formalism and thus yielding a good theory of motivic cohomology. His construction is a bit indirect and it would be nice to have a direct construction of a representing motivic ring spectrum H Z S S H ( S ) H Z S ∈ S H ( S ) HZ_(S)inSH(S)H \mathbb{Z}_{S} \in \mathrm{SH}(S)HZS∈SH(S) for each base-scheme S S SSS, still satisfying the cartesian condition.
Hoyois has constructed such a theory of motivic cohomology over an arbitrary basescheme by using a recent breakthrough in our understanding of the motivic stable homotopy categories S H ( S ) S H ( S ) SH(S)\mathrm{SH}(S)SH(S). This is a new construction of S H ( S ) S H ( S ) SH(S)\mathrm{SH}(S)SH(S) more in line with Voevodsky construction of D M ( k ) D M ( k ) DM(k)\mathrm{DM}(k)DM(k). The basic idea is sketched in notes of Voevodsky [126], which were realized in a series of works by Ananyevskiy, Garkusha, Panin, Neshitov [2,4,45-48](authorship in various combinations). Building on these works, Elmanto, Hoyois, Khan, Sosnilo, and Yakerson [36-38] construct an infinity category of framed correspondences, and use the basic program of Voevodsky's construction of D M ( k ) D M ( k ) DM(k)\mathrm{DM}(k)DM(k) to realize S H ( S ) S H ( S ) SH(S)\mathrm{SH}(S)SH(S) as arising from presheaves of spectra with framed transfers, just as objects of D M ( k ) D M ( k ) DM(k)\mathrm{DM}(k)DM(k) arise from presheaves of complexes of sheaves with transfers for finite correspondences. It is not our purpose here to give a detailed discussion of this beautiful topic; we content ourselves with sketching some of the basic principles.
An integral closed subscheme Z X × Y Z ⊂ X × Y Z sub X xx YZ \subset X \times YZ⊂X×Y that defines a finite correspondence from X X XXX to Y Y YYY can be thought of a special type of a span via the two projections
For X X XXX and Y Y YYY smooth and finite type over a given base-scheme S S SSS, a framed correspondence from X X XXX to Y Y YYY is also a span,
satisfying certain conditions, together with some additional data (the framing). For simplicity, assume that X X XXX is connected. The morphism p p ppp is required to be a finite, flat, local complete intersection (lci) morphism, called a finite syntomic morphism (the terminology was introduced by Mazur). The lci condition means that p p ppp factors as closed immersion i : Z P i : Z → P i:Z rarr Pi: Z \rightarrow Pi:Z→P followed by a smooth morphism f : P X f : P → X f:P rarr Xf: P \rightarrow Xf:P→X, and the closed subscheme i ( Z ) i ( Z ) i(Z)i(Z)i(Z) of P P PPP is locally defined by exactly dim X P dim X Z dim X ⁡ P − dim X ⁡ Z dim_(X)P-dim_(X)Z\operatorname{dim}_{X} P-\operatorname{dim}_{X} ZdimX⁡P−dimX⁡Z equations forming a regular sequence. The morphism p p ppp factored in this way has a relative cotangent complex L p L p L_(p)\mathbb{L}_{p}Lp admitting a simple description, namely
L p = [ l Z / l Z 2 d i Ω P / X ] L p = l Z / l Z 2 → d i ∗ Ω P / X L_(p)=[l_(Z)//l_(Z)^(2)rarr"d"i^(**)Omega_(P//X)]\mathbb{L}_{p}=\left[\mathscr{l}_{Z} / \mathscr{l}_{Z}^{2} \xrightarrow{d} i^{*} \Omega_{P / X}\right]Lp=[lZ/lZ2→di∗ΩP/X]
the conditions on i i iii and p p ppp say that both Z / Z 2 â„“ Z / â„“ Z 2 â„“_(Z)//â„“_(Z)^(2)\ell_{Z} / \ell_{Z}^{2}â„“Z/â„“Z2 and i Ω P / X i ∗ Ω P / X i^(**)Omega_(P//X)i^{*} \Omega_{P / X}i∗ΩP/X are locally free coherent sheaves on Z Z ZZZ of rank dim X P dim X Z rank ⁡ dim X ⁡ P − dim X ⁡ Z rank dim_(X)P-dim_(X)Z\operatorname{rank} \operatorname{dim}_{X} P-\operatorname{dim}_{X} Zrank⁡dimX⁡P−dimX⁡Z and dim X P dim X ⁡ P dim_(X)P\operatorname{dim}_{X} PdimX⁡P, respectively. For p p ppp an lci morphism, the perfect
complex L p L p L_(p)\mathbb{L}_{p}Lp defines a point { L p } L p {L_(p)}\left\{\mathbb{L}_{p}\right\}{Lp} in the space K ( Z ) K ( Z ) K(Z)\mathcal{K}(Z)K(Z) defining the K K KKK-theory of Z Z ZZZ; in the case of a finite syntomic morphism, the virtual rank of { L p } L p {L_(p)}\left\{\mathbb{L}_{p}\right\}{Lp} is zero.
A framing for a syntomic map p : Z X p : Z → X p:Z rarr Xp: Z \rightarrow Xp:Z→X is a choice of a path γ : [ 0 , 1 ] K ( Z ) γ : [ 0 , 1 ] → K ( Z ) gamma:[0,1]rarrK(Z)\gamma:[0,1] \rightarrow \mathcal{K}(Z)γ:[0,1]→K(Z) connecting { L p } L p {L_(p)}\left\{\mathbb{L}_{p}\right\}{Lp} with the base-point 0 K ( Z ) 0 ∈ K ( Z ) 0inK(Z)0 \in \mathcal{K}(Z)0∈K(Z). For a framing to exist, the class [ L p ] K 0 ( Z ) L p ∈ K 0 ( Z ) [L_(p)]inK_(0)(Z)\left[\mathbb{L}_{p}\right] \in K_{0}(Z)[Lp]∈K0(Z) must be zero, but the choice of γ γ gamma\gammaγ is additional data. The morphism q : Z Y q : Z → Y q:Z rarr Yq: Z \rightarrow Yq:Z→Y is arbitrary.
One has the usual notion of a composition of spans:
which preserves the finite syntomic condition. However, one needs a higher categorical structure to take care of associativity constraints. The composition of paths is even trickier, since we are dealing here with actual paths, not paths up to homotopy. In the end, this produces an infinity category Corr f r ( S m S ) f r S m S ^(fr)(Sm_(S)){ }^{\mathrm{fr}}\left(\mathrm{Sm}_{S}\right)fr(SmS) of framed correspondences on smooth S S SSS-schemes, rather than a category; roughly speaking, the composition is only defined "up to homotopy and coherent higher homotopies."
Via the infinity category Corr f r ( S m S ) Corr f r ⁡ S m S Corr^(fr)(Sm_(S))\operatorname{Corr}^{\mathrm{fr}}\left(\mathrm{Sm}_{S}\right)Corrfr⁡(SmS), we have the infinity category of framed motivic spaces, H f r ( S ) H f r ( S ) H^(fr)(S)\mathbf{H}^{\mathrm{fr}}(S)Hfr(S), this being the infinity category of A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant, Nisnevich sheaves of spaces on C o r r f r ( S m S ) C o r r f r S m S Corr^(fr)(Sm_(S))\mathbf{C o r r}{ }^{\mathrm{fr}}\left(\mathrm{Sm}_{S}\right)Corrfr(SmS). There is a stable version, S H f r ( S ) S H f r ( S ) SH^(fr)(S)\mathbf{S H}^{\mathrm{fr}}(S)SHfr(S), an infinite suspension functor Σ f r : H f r ( S ) S H f r ( S ) Σ f r ∞ : H f r ( S ) → S H f r ( S ) Sigma_(fr)^(oo):H^(fr)(S)rarrSH^(fr)(S)\Sigma_{\mathrm{fr}}^{\infty}: \mathbf{H}^{\mathrm{fr}}(S) \rightarrow \mathbf{S H}^{\mathrm{fr}}(S)Σfr∞:Hfr(S)→SHfr(S), and an equivalence of infinity categories γ : S H f r ( S ) S H ( S ) γ ∗ : S H f r ( S ) → S H ( S ) gamma_(**):SH^(fr)(S)rarrSH(S)\gamma_{*}: \mathbf{S H}^{\mathrm{fr}}(S) \rightarrow \mathbf{S H}(S)γ∗:SHfr(S)→SH(S), where S H ( S ) S H ( S ) SH(S)\mathbf{S H}(S)SH(S) is the infinity category version of the triangulated category S H ( S ) S H ( S ) SH(S)\mathrm{SH}(S)SH(S), that is, the homotopy category of S H ( S ) S H ( S ) SH(S)\mathbf{S H}(S)SH(S) is SH ( S ) SH ⁡ ( S ) SH(S)\operatorname{SH}(S)SH⁡(S). The equivalence γ γ ∗ gamma_(**)\gamma_{*}γ∗ can be thought of as a version of the construction of infinite loop spaces from Segal's Γ Î“ Gamma\GammaΓ-spaces, with a framed correspondence X Z Y X ← Z → Y X larr Z rarr YX \leftarrow Z \rightarrow YX←Z→Y of degree n n nnn over X X XXX being viewed as a generalization of the map [ n ] + [ 0 ] + [ n ] + → [ 0 ] + [n]_(+)rarr[0]_(+)[n]_{+} \rightarrow[0]_{+}[n]+→[0]+ in Γ op Γ op  Gamma^("op ")\Gamma^{\text {op }}Γop .
With this background, we can give a rough idea of Hoyois' construction of the spectrum representing motivic cohomology over S S SSS in [63]. He considers spans X p Z q Y X ← p Z → q Y Xlarr^(p)Zrarr"q"YX \stackrel{p}{\leftarrow} Z \xrightarrow{q} YX←pZ→qY, X , Y Sm S X , Y ∈ Sm S X,Y inSm_(S)X, Y \in \operatorname{Sm}_{S}X,Y∈SmS, with p : Z X p : Z → X p:Z rarr Xp: Z \rightarrow Xp:Z→X a finite morphism such that p O Z p ∗ O Z p_(**)O_(Z)p_{*} \mathcal{O}_{Z}p∗OZ is a locally free O X O X − O_(X^(-))\mathcal{O}_{X^{-}}OX− module; note that this condition is satisfied if p p ppp is a syntomic morphism, but not conversely. These spans form a category Corr flf ( S m S ) flf  S m S ^("flf ")(Sm_(S)){ }^{\text {flf }}\left(\mathrm{Sm}_{S}\right)flf (SmS) under span composition ("flf" stands for "finite, locally free") and forgetting the paths γ γ gamma\gammaγ defines a morphism of (infinity) categories π a d : Corr f r ( S m S ) Corr f l f ( S m S ) Ï€ a d : Corr f r ⁡ S m S → Corr f l f ⁡ S m S pi_(ad):Corr^(fr)(Sm_(S))rarrCorr^(flf)(Sm_(S))\pi_{\mathrm{ad}}: \operatorname{Corr}^{\mathrm{fr}}\left(\mathrm{Sm}_{S}\right) \rightarrow \operatorname{Corr}^{\mathrm{flf}}\left(\mathrm{Sm}_{S}\right)Ï€ad:Corrfr⁡(SmS)→Corrflf⁡(SmS).
Given a commutative monoid A A AAA, the constant Nisnevich sheaf on S m S S m S Sm_(S)\mathrm{Sm}_{S}SmS with value A A AAA extends to a functor
A S : ( C o r r f l f ) o p A b A S : C o r r f l f o p → A b A_(S):(Corr^(flf))^(op)rarrAbA_{S}:\left(\mathbf{C o r r}^{\mathrm{flf}}\right)^{\mathrm{op}} \rightarrow \mathbf{A b}AS:(Corrflf)op→Ab
where the pullback from Y Y YYY to X X XXX by X p Z q Y X ← p Z → q Y Xlarr^(p)Zrarr"q"YX \stackrel{p}{\leftarrow} Z \xrightarrow{q} YX←pZ→qY is given by multiplication by r n k O X O Z r n k O X O Z rnk_(O_(X))O_(Z)\mathrm{rnk}_{\mathcal{O}_{X}} \mathcal{O}_{Z}rnkOXOZ if X X XXX and Y Y YYY are connected; one extends to general smooth X X XXX and Y Y YYY by additivity. This gives us the presheaf (of abelian monoids) with framed transfers A S f r := A S π a d o p A S f r := A S ∘ Ï€ a d o p A_(S)^(fr):=A_(S)@pi_(ad)^(op)A_{S}^{\mathrm{fr}}:=A_{S} \circ \pi_{\mathrm{ad}}^{\mathrm{op}}ASfr:=AS∘πadop, and the machinery of [36-38] converts this into the motivic spectrum γ Σ f r A S f r SH ( S ) γ ∗ Σ f r ∞ A S f r ∈ SH ⁡ ( S ) gamma_(**)Sigma_(fr)^(oo)A_(S)^(fr)in SH(S)\gamma_{*} \Sigma_{\mathrm{fr}}^{\infty} A_{S}^{\mathrm{fr}} \in \operatorname{SH}(S)γ∗Σfr∞ASfr∈SH⁡(S). Hoyois shows [63,
LEMMA 20] that this construction produces a cartesian family, and that taking A = Z A = Z A=ZA=\mathbb{Z}A=Z recovers Spitzweck's family S M Z S S ↦ M Z S S|->MZ_(S)S \mapsto M \mathbb{Z}_{S}S↦MZS [63, THEOREM 21].
This gives us a conceptually simple construction of a motivic Eilenberg-MacLane spectrum, and the corresponding motivic category D M H ( S ) D M H ( S ) DM_(H)(S)\mathrm{DM}_{H}(S)DMH(S), much in the spirit of Voevodsky original construction of D M ( k ) D M ( k ) DM(k)\mathrm{DM}(k)DM(k) and the Röndigs- stvær theorem identifying D M ( k ) D M ( k ) DM(k)\mathrm{DM}(k)DM(k) with the homotopy category of EM ( Z ( 0 ) ) EM ⁡ ( Z ( 0 ) ) EM(Z(0))\operatorname{EM}(\mathbb{Z}(0))EM⁡(Z(0))-modules.

4. MILNOR-WITT MOTIVIC COHOMOLOGY

The classical Chow group C H n ( X ) C H n ( X ) CH^(n)(X)\mathrm{CH}^{n}(X)CHn(X) of codimension n n nnn algebraic cycles modulo rational equivalence on a smooth variety X X XXX is part of the motivic cohomology of X X XXX via the isomorphism C H n ( X ) = H 2 n ( X , Z ( n ) ) C H n ( X ) = H 2 n ( X , Z ( n ) ) CH^(n)(X)=H^(2n)(X,Z(n))\mathrm{CH}^{n}(X)=H^{2 n}(X, \mathbb{Z}(n))CHn(X)=H2n(X,Z(n)). Barge and Morel [12] have introduced a refinement of the Chow groups, the Chow-Witt groups, that incorporates information about quadratic forms. Their construction has been embedded in a larger theory of Milnor-Witt motives and Milnor-Witt motivic cohomology, which we describe in this section. The quadratic information given by the Chow-Witt groups, Milnor-Witt motivic cohomology and related theories has proven useful in recent efforts to give quadratic refinements for intersection theory and enumerative geometry; see [ 10 , 11 , 21 , 61 , 76 , 77 , 86 ] [ 10 , 11 , 21 , 61 , 76 , 77 , 86 ] [10,11,21,61,76,77,86][10,11,21,61,76,77,86][10,11,21,61,76,77,86] for some examples. We refer the reader to [ 8 , 31 , 39 , 92 ] [ 8 , 31 , 39 , 92 ] [8,31,39,92][8,31,39,92][8,31,39,92] for details on the theory described in this section.

4.1. Milnor-Witt K K KKK-theory and the Chow-Witt groups

A codimension n n nnn algebraic cycle Z := i n i Z i Z := ∑ i   n i Z i Z:=sum_(i)n_(i)Z_(i)Z:=\sum_{i} n_{i} Z_{i}Z:=∑iniZi can be thought of as the set of its generic points z i z i z_(i)z_{i}zi together with the Z Z Z\mathbb{Z}Z-valued function n i n i n_(i)n_{i}ni on z i z i z_(i)z_{i}zi, from which we can write the group Z n ( X ) Z n ( X ) Z^(n)(X)Z^{n}(X)Zn(X) of codimension n n nnn algebraic cycles as
Z n ( X ) = z X ( n ) Z Z n ( X ) = ⨁ z ∈ X ( n )   Z Z^(n)(X)=bigoplus_(z inX^((n)))ZZ^{n}(X)=\bigoplus_{z \in X^{(n)}} \mathbb{Z}Zn(X)=⨁z∈X(n)Z
where X ( n ) X ( n ) X^((n))X^{(n)}X(n) is the set of points z X z ∈ X z in Xz \in Xz∈X with closure Z := z ¯ X Z := z ¯ ⊂ X Z:= bar(z)sub XZ:=\bar{z} \subset XZ:=z¯⊂X of codimension n n nnn.
Let G W ( F ) G W ( F ) GW(F)\mathrm{GW}(F)GW(F) denote the Grothendieck-Witt ring of virtual non-degenerate quadratic forms over F F FFF and let W ( F ) = GW ( F ) / ( H ) W ( F ) = GW ⁡ ( F ) / ( H ) W(F)=GW(F)//(H)W(F)=\operatorname{GW}(F) /(H)W(F)=GW⁡(F)/(H) where H H HHH is the hyperbolic form H ( x , y ) = H ( x , y ) = H(x,y)=H(x, y)=H(x,y)= x 2 y 2 x 2 − y 2 x^(2)-y^(2)x^{2}-y^{2}x2−y2 (we assume throughout that the characteristic is 2 ≠ 2 !=2\neq 2≠2 to avoid technical difficulties); W ( F ) W ( F ) W(F)W(F)W(F) is the Witt ring of anisotropic quadratic forms over F F FFF (see [107]).
One can consider a finite set of codimension n n nnn points z i X ( n ) z i ∈ X ( n ) z_(i)inX^((n))z_{i} \in X^{(n)}zi∈X(n), together with a collection of classes { q i G W ( k ( z i ) ) } q i ∈ G W k z i {q_(i)inGW(k(z_(i)))}\left\{q_{i} \in \mathrm{GW}\left(k\left(z_{i}\right)\right)\right\}{qi∈GW(k(zi))}; one recovers a Z Z Z\mathbb{Z}Z-valued function on z i z i z_(i)z_{i}zi by taking the rank of q i q i q_(i)q_{i}qi. This gives the group
Z ~ n ( X ) := z X ( n ) G W ( k ( z ) ) Z ~ n ( X ) := z ∈ X ( n ) G W ( k ( z ) ) tilde(Z)^(n)(X):=_(z inX^((n)))GW(k(z))\tilde{Z}^{n}(X):=\underset{z \in X^{(n)}}{ } \mathrm{GW}(k(z))Z~n(X):=z∈X(n)GW(k(z))
with rank homomorphism rnk: Z ~ n ( X ) Z n ( X ) Z ~ n ( X ) → Z n ( X ) tilde(Z)^(n)(X)rarrZ^(n)(X)\tilde{Z}^{n}(X) \rightarrow Z^{n}(X)Z~n(X)→Zn(X). In contrast with integer-valued functions, an element q G W ( k ( z ) ) q ∈ G W ( k ( z ) ) q inGW(k(z))q \in \mathrm{GW}(k(z))q∈GW(k(z)) does not always extend to all of z ¯ z ¯ bar(z)\bar{z}z¯; there is an obstruction given by a certain boundary map
: GW ( k ( z ) ) w z ¯ X ( n + 1 ) W ( k ( w ) ) ∂ : GW ⁡ ( k ( z ) ) → ⨁ w ∈ z ¯ ∩ X ( n + 1 )   W ( k ( w ) ) del:GW(k(z))rarrbigoplus_(w in bar(z)nnX^((n+1)))W(k(w))\partial: \operatorname{GW}(k(z)) \rightarrow \bigoplus_{w \in \bar{z} \cap X^{(n+1)}} W(k(w))∂:GW⁡(k(z))→⨁w∈z¯∩X(n+1)W(k(w))
This starts to look more like classical homology, in that one should consider Z ~ n ( X ) Z ~ n ( X ) tilde(Z)^(n)(X)\tilde{Z}^{n}(X)Z~n(X) as a group of chains rather than a group of cycles.
This is not enough, as one needs a quadratic refinement for the classical relation given by rational equivalence. The original construction of Barge-Morel defined this relation, but later developments put their construction in a rather more natural form, which we now describe.
We recall that the Milnor K K KKK-theory ring K M ( F ) := n 0 K n M ( F ) K ∗ M ( F ) := ⨁ n ≥ 0   K n M ( F ) K_(**)^(M)(F):=bigoplus_(n >= 0)K_(n)^(M)(F)K_{*}^{M}(F):=\bigoplus_{n \geq 0} K_{n}^{M}(F)K∗M(F):=⨁n≥0KnM(F) of a field F F FFF is defined as the quotient of the tensor algebra on the abelian group of units F × F × F^(xx)F^{\times}F×, modulo the Steinberg relation
The quadratic refinement of K M ( F ) K ∗ M ( F ) K_(**)^(M)(F)K_{*}^{M}(F)K∗M(F) is the Hopkins-Morel Milnor-Witt K K KKK-theory of F F FFF.
Definition 4.1 (Hopkins-Morel [92, Definition 6.3.1]). Let F F FFF be a field. The Milnor-Witt K K KKK-theory of F , K M W ( F ) := n Z K n M W ( F ) F , K ∗ M W ( F ) := ⨁ n ∈ Z   K n M W ( F ) F,K_(**)^(MW)(F):=bigoplus_(n inZ)K_(n)^(MW)(F)F, K_{*}^{\mathrm{MW}}(F):=\bigoplus_{n \in \mathbb{Z}} K_{n}^{\mathrm{MW}}(F)F,K∗MW(F):=⨁n∈ZKnMW(F), is the Z Z Z\mathbb{Z}Z-graded associative algebra defined by the following generators and relations.

Generators

(G1) For each u F × u ∈ F × u inF^(xx)u \in F^{\times}u∈F×, we have the generator [ u ] [ u ] [u][u][u] of degree 1 ;
(G2) There is an additional generator η η eta\etaη of degree -1 .

Relations

(R0) η [ u ] = [ u ] η η â‹… [ u ] = [ u ] â‹… η eta*[u]=[u]*eta\eta \cdot[u]=[u] \cdot \etaη⋅[u]=[u]⋅η;
(R1) [ u v ] = [ u ] + [ v ] + η [ u ] [ v ] [ u v ] = [ u ] + [ v ] + η â‹… [ u ] â‹… [ v ] [uv]=[u]+[v]+eta*[u]*[v][u v]=[u]+[v]+\eta \cdot[u] \cdot[v][uv]=[u]+[v]+η⋅[u]â‹…[v];
(R2) [ u ] [ 1 u ] = 0 [ u ] ⋅ [ 1 − u ] = 0 [u]*[1-u]=0[u] \cdot[1-u]=0[u]⋅[1−u]=0 for u F { 0 , 1 } u ∈ F ∖ { 0 , 1 } u in F\\{0,1}u \in F \backslash\{0,1\}u∈F∖{0,1};
(R3) Let h = ( 2 + η [ 1 ] ) h = ( 2 + η â‹… [ − 1 ] ) h=(2+eta*[-1])h=(2+\eta \cdot[-1])h=(2+η⋅[−1]). Then η h = 0 η â‹… h = 0 eta*h=0\eta \cdot h=0η⋅h=0.
It follows directly that sending [ u ] [ u ] [u][u][u] to { u } K 1 M ( F ) { u } ∈ K 1 M ( F ) {u}inK_(1)^(M)(F)\{u\} \in K_{1}^{M}(F){u}∈K1M(F) and sending η η eta\etaη to zero defines a surjective graded algebra homomorphism K M W ( F ) K M ( F ) K ∗ M W ( F ) → K ∗ M ( F ) K_(**)^(MW)(F)rarrK_(**)^(M)(F)K_{*}^{M W}(F) \rightarrow K_{*}^{M}(F)K∗MW(F)→K∗M(F) with kernel ( η ) ( η ) (eta)(\eta)(η). We write [ u 1 , , u n ] u 1 , … , u n [u_(1),dots,u_(n)]\left[u_{1}, \ldots, u_{n}\right][u1,…,un] for the product [ u 1 ] [ u n ] u 1 ⋯ u n [u_(1)]cdots[u_(n)]\left[u_{1}\right] \cdots\left[u_{n}\right][u1]⋯[un].
Theorem 4.2 (Hopkins-Morel [92, THEOREM 6.4.5]). Let I ( F ) G W ( F ) I ( F ) ⊂ G W ( F ) I(F)subGW(F)I(F) \subset \mathrm{GW}(F)I(F)⊂GW(F) be the kernel of the rank homomorphism G W ( F ) Z G W ( F ) → Z GW(F)rarrZ\mathrm{GW}(F) \rightarrow \mathbb{Z}GW(F)→Z, with the nth power ideal I n ( F ) G W ( F ) I n ( F ) ⊂ G W ( F ) I^(n)(F)subGW(F)I^{n}(F) \subset \mathrm{GW}(F)In(F)⊂GW(F) for n > 0 n > 0 n > 0n>0n>0. Define I n ( F ) = W ( F ) I n ( F ) = W ( F ) I^(n)(F)=W(F)I^{n}(F)=W(F)In(F)=W(F) for n 0 n ≤ 0 n <= 0n \leq 0n≤0. Then for each n Z n ∈ Z n inZn \in \mathbb{Z}n∈Z, the surjection K n M W ( F ) K n M ( F ) K n M W ( F ) → K n M ( F ) K_(n)^(MW)(F)rarrK_(n)^(M)(F)K_{n}^{\mathrm{MW}}(F) \rightarrow K_{n}^{M}(F)KnMW(F)→KnM(F) extends to an exact sequence
0 I n + 1 ( F ) K n M W ( F ) K n M ( F ) 0 0 → I n + 1 ( F ) → K n M W ( F ) → K n M ( F ) → 0 0rarrI^(n+1)(F)rarrK_(n)^(MW)(F)rarrK_(n)^(M)(F)rarr00 \rightarrow I^{n+1}(F) \rightarrow K_{n}^{\mathrm{MW}}(F) \rightarrow K_{n}^{M}(F) \rightarrow 00→In+1(F)→KnMW(F)→KnM(F)→0
For n = 0 , K 0 M ( F ) = Z , K 0 M W ( F ) n = 0 , K 0 M ( F ) = Z , K 0 M W ( F ) n=0,K_(0)^(M)(F)=Z,K_(0)^(MW)(F)n=0, K_{0}^{M}(F)=\mathbb{Z}, K_{0}^{\mathrm{MW}}(F)n=0,K0M(F)=Z,K0MW(F) is isomorphic to G W ( F ) G W ( F ) GW(F)\mathrm{GW}(F)GW(F) and the above sequence is isomorphic to the defining sequence for I ( F ) I ( F ) I(F)I(F)I(F). For n < 0 , K n M ( F ) = 0 n < 0 , K n M ( F ) = 0 n < 0,K_(n)^(M)(F)=0n<0, K_{n}^{M}(F)=0n<0,KnM(F)=0 and K n M W ( F ) K n M W ( F ) ≅ K_(n)^(MW)(F)~=K_{n}^{\mathrm{MW}}(F) \congKnMW(F)≅
W ( F ) W ( F ) W(F)W(F)W(F). Finally, we have, for each n < 0 n < 0 n < 0n<0n<0, a commutative diagram
and, for n = 0 n = 0 n=0n=0n=0, the commutative diagram
where π Ï€ pi\piÏ€ is the canonical surjection.
The isomorphism G W ( F ) K 0 M W ( F ) G W ( F ) → ∼ K 0 M W ( F ) GW(F)rarr"∼"K_(0)^(MW)(F)\mathrm{GW}(F) \xrightarrow{\sim} K_{0}^{\mathrm{MW}}(F)GW(F)→∼K0MW(F) sends u ⟨ u ⟩ (:u:)\langle u\rangle⟨u⟩ to 1 + η [ u ] 1 + η [ u ] 1+eta[u]1+\eta[u]1+η[u], where u ⟨ u ⟩ (:u:)\langle u\rangle⟨u⟩ is the rank one form u ( x ) := u x 2 ⟨ u ⟩ ( x ) := u x 2 (:u:)(x):=ux^(2)\langle u\rangle(x):=u x^{2}⟨u⟩(x):=ux2; since a quadratic form over F F FFF is diagonalizable (char F 2 F ≠ 2 F!=2F \neq 2F≠2 ), the isomorphism is completely determined by its value on the forms u ⟨ u ⟩ (:u:)\langle u\rangle⟨u⟩. Given a 1-dimensional F F FFF-vector space L L LLL, we have the GW ( F ) GW ⁡ ( F ) GW(F)\operatorname{GW}(F)GW⁡(F)-module GW ( F ; L ) GW ⁡ ( F ; L ) GW(F;L)\operatorname{GW}(F ; L)GW⁡(F;L) of non-degenerate, L L LLL-valued quadratic forms q : V L q : V → L q:V rarr Lq: V \rightarrow Lq:V→L; each vector space isomorphism ϕ : L F Ï• : L → F phi:L rarr F\phi: L \rightarrow FÏ•:L→F gives an isomorphism of GW ( F ) GW ⁡ ( F ) GW(F)\operatorname{GW}(F)GW⁡(F)-modules GW ( F ; L ) GW ( F ) GW ⁡ ( F ; L ) ≅ GW ⁡ ( F ) GW(F;L)~=GW(F)\operatorname{GW}(F ; L) \cong \operatorname{GW}(F)GW⁡(F;L)≅GW⁡(F). Since K M W ( F ) K ∗ M W ( F ) K_(**)^(MW)(F)K_{*}^{\mathrm{MW}}(F)K∗MW(F) is a Z Z Z\mathbb{Z}Z-graded K 0 M W ( F ) = G W ( F ) K 0 M W ( F ) = G W ( F ) K_(0)^(MW)(F)=GW(F)K_{0}^{\mathrm{MW}}(F)=\mathrm{GW}(F)K0MW(F)=GW(F)-module, we can form the Z Z Z\mathbb{Z}Z-graded K M W ( F ) K ∗ M W ( F ) K_(**)^(MW)(F)K_{*}^{\mathrm{MW}}(F)K∗MW(F)-module K M W ( F ; L ) := K ∗ M W ( F ; L ) := K_(**)^(MW)(F;L):=K_{*}^{\mathrm{MW}}(F ; L):=K∗MW(F;L):= GW ( F ; L ) G W ( F ) K M W ( F ) GW ⁡ ( F ; L ) ⊗ G W ( F ) K ∗ M W ( F ) GW(F;L)ox_(GW(F))K_(**)^(MW)(F)\operatorname{GW}(F ; L) \otimes_{\mathrm{GW}(F)} K_{*}^{\mathrm{MW}}(F)GW⁡(F;L)⊗GW(F)K∗MW(F).
Given a dvr O dvr ⁡ O dvr O\operatorname{dvr} \mathcal{O}dvr⁡O with residue field k k kkk, quotient field F F FFF, and generator t t ttt for the maximal ideal, one has the map
t : K n M W ( F ) K n 1 M W ( k ) ∂ t : K n M W ( F ) → K n − 1 M W ( k ) del_(t):K_(n)^(MW)(F)rarrK_(n-1)^(MW)(k)\partial_{t}: K_{n}^{\mathrm{MW}}(F) \rightarrow K_{n-1}^{\mathrm{MW}}(k)∂t:KnMW(F)→Kn−1MW(k)
determined by the formulas
t ( [ t , u 2 , , u n ] ) = [ u ¯ 2 , , u ¯ n ] , t ( [ u 1 , u 2 , , u n ] ) = 0 , t ( η x ) = η t ( x ) ∂ t t , u 2 , … , u n = u ¯ 2 , … , u ¯ n , ∂ t u 1 , u 2 , … , u n = 0 , ∂ t ( η â‹… x ) = η â‹… ∂ t ( x ) del_(t)([t,u_(2),dots,u_(n)])=[ bar(u)_(2),dots, bar(u)_(n)],quaddel_(t)([u_(1),u_(2),dots,u_(n)])=0,quaddel_(t)(eta*x)=eta*del_(t)(x)\partial_{t}\left(\left[t, u_{2}, \ldots, u_{n}\right]\right)=\left[\bar{u}_{2}, \ldots, \bar{u}_{n}\right], \quad \partial_{t}\left(\left[u_{1}, u_{2}, \ldots, u_{n}\right]\right)=0, \quad \partial_{t}(\eta \cdot x)=\eta \cdot \partial_{t}(x)∂t([t,u2,…,un])=[u¯2,…,u¯n],∂t([u1,u2,…,un])=0,∂t(η⋅x)=η⋅∂t(x)
for u 1 , , u n O × u 1 , … , u n ∈ O × u_(1),dots,u_(n)inO^(xx)u_{1}, \ldots, u_{n} \in \mathcal{O}^{\times}u1,…,un∈O×, and x K n + 1 M W ( F ) x ∈ K n + 1 M W ( F ) x inK_(n+1)^(MW)(F)x \in K_{n+1}^{\mathrm{MW}}(F)x∈Kn+1MW(F), where u ¯ i u ¯ i bar(u)_(i)\bar{u}_{i}u¯i is the image of u i u i u_(i)u_{i}ui in k × k × k^(xx)k^{\times}k×. This is similar to the well-known boundary map : K n M ( F ) K n 1 M ( k ) ∂ : K n M ( F ) → K n − 1 M ( k ) del:K_(n)^(M)(F)rarrK_(n-1)^(M)(k)\partial: K_{n}^{M}(F) \rightarrow K_{n-1}^{M}(k)∂:KnM(F)→Kn−1M(k), with the difference, that ∂ del\partial∂ does not depend on the choice of t t ttt while t ∂ t del_(t)\partial_{t}∂t does. To get a boundary map that is independent of the choice of parameter t t ttt, one needs to include the twisting. This yields the well-defined boundary map
: K n M W ( F ; L O F ) K n 1 M W ( k ; L O ( m / m 2 ) ) ∂ : K n M W F ; L ⊗ O F → K n − 1 M W k ; L ⊗ O m / m 2 ∨ del:K_(n)^(MW)(F;Lox_(O)F)rarrK_(n-1)^(MW)(k;Lox_(O)(m//m^(2))^(vv))\partial: K_{n}^{\mathrm{MW}}\left(F ; L \otimes_{\mathcal{O}} F\right) \rightarrow K_{n-1}^{\mathrm{MW}}\left(k ; L \otimes_{\mathcal{O}}\left(\mathfrak{m} / \mathfrak{m}^{2}\right)^{\vee}\right)∂:KnMW(F;L⊗OF)→Kn−1MW(k;L⊗O(m/m2)∨)
for L L LLL a free rank-one O O O\mathcal{O}O-module, independent of the choice of generator for the maximal ideal m m m\mathfrak{m}m, where ∂ del\partial∂ is defined by choosing a generator t t ttt and an O O O\mathcal{O}O-basis λ λ lambda\lambdaλ for L L LLL, and setting
( x λ ) := t ( x ) λ / t ∂ ( x ⊗ λ ) := ∂ t ( x ) ⊗ λ ⊗ ∂ / ∂ t del(x ox lambda):=del_(t)(x)ox lambda ox del//del t\partial(x \otimes \lambda):=\partial_{t}(x) \otimes \lambda \otimes \partial / \partial t∂(x⊗λ):=∂t(x)⊗λ⊗∂/∂t
Definition 4.3. Let X X XXX be a smooth finite type k k kkk-scheme, and let L L L\mathscr{L}L be an invertible sheaf on X X XXX. The n n nnnth L L L\mathscr{L}L-twisted Rost-Schmid complex for Milnor-Witt K K KKK-theory is the complex RS ( X , L , n ) RS ∗ ⁡ ( X , L , n ) RS^(**)(X,L,n)\operatorname{RS}^{*}(X, \mathscr{L}, n)RS∗⁡(X,L,n) with
RS m ( X , L , n ) := x X ( m ) K n m M W ( k ( x ) ; L x O X , x m ( m x / m x 2 ) ) RS m ⁡ ( X , L , n ) := ⨁ x ∈ X ( m )   K n − m M W k ( x ) ; L x ⊗ O X , x ⋀ m   m x / m x 2 ∨ RS^(m)(X,L,n):=bigoplus_(x inX^((m)))K_(n-m)^(MW)(k(x);L_(x)ox_(O_(X,x))^^^m(m_(x)//m_(x)^(2))^(vv))\operatorname{RS}^{m}(X, \mathscr{L}, n):=\bigoplus_{x \in X^{(m)}} K_{n-m}^{\mathrm{MW}}\left(k(x) ; \mathscr{L}_{x} \otimes_{\mathcal{O}_{X, x}} \bigwedge^{m}\left(\mathfrak{m}_{x} / \mathfrak{m}_{x}^{2}\right)^{\vee}\right)RSm⁡(X,L,n):=⨁x∈X(m)Kn−mMW(k(x);Lx⊗OX,x⋀m(mx/mx2)∨)
and boundary map m : RS m ( X , L , n ) RS m + 1 ( X , L , n ) ∂ m : RS m ⁡ ( X , L , n ) → RS m + 1 ⁡ ( X , L , n ) del^(m):RS^(m)(X,L,n)rarrRS^(m+1)(X,L,n)\partial^{m}: \operatorname{RS}^{m}(X, \mathscr{L}, n) \rightarrow \operatorname{RS}^{m+1}(X, \mathscr{L}, n)∂m:RSm⁡(X,L,n)→RSm+1⁡(X,L,n) the sum of the maps
w , x : K n m M W ( k ( x ) ; L x O X , x m ( m x / m x 2 ) ) K n m 1 M W ( k ( w ) ; L x X , x m + 1 ( m w / m w 2 ) ) ∂ w , x : K n − m M W k ( x ) ; L x ⊗ O X , x ⋀ m   m x / m x 2 ∨ → K n − m − 1 M W k ( w ) ; L x ⊗ ⊗ X , x ⋀ m + 1   m w / m w 2 ∨ {:[del_(w,x):K_(n-m)^(MW)(k(x);L_(x)oxO_(X,x)^^^m(m_(x)//m_(x)^(2))^(vv))],[ rarrK_(n-m-1)^(MW)(k(w);L_(x)oxox_(X,x)^^^m+1(m_(w)//m_(w)^(2))^(vv))]:}\begin{aligned} \partial_{w, x} & : K_{n-m}^{\mathrm{MW}}\left(k(x) ; \mathscr{L}_{x} \otimes \mathcal{O}_{X, x} \bigwedge^{m}\left(\mathfrak{m}_{x} / \mathfrak{m}_{x}^{2}\right)^{\vee}\right) \\ & \rightarrow K_{n-m-1}^{\mathrm{MW}}\left(k(w) ; \mathscr{L}_{x} \otimes \otimes_{X, x} \bigwedge^{m+1}\left(\mathfrak{m}_{w} / \mathfrak{m}_{w}^{2}\right)^{\vee}\right) \end{aligned}∂w,x:Kn−mMW(k(x);Lx⊗OX,x⋀m(mx/mx2)∨)→Kn−m−1MW(k(w);Lx⊗⊗X,x⋀m+1(mw/mw2)∨)
associated to the normalization of the local ring O x ¯ , w O x ¯ , w O_( bar(x),w)\mathcal{O}_{\bar{x}, w}Ox¯,w for w x ¯ X ( m + 1 ) w ∈ x ¯ ∩ X ( m + 1 ) w in bar(x)nnX^((m+1))w \in \bar{x} \cap X^{(m+1)}w∈x¯∩X(m+1). Here we have cheated a bit in the definition of w , x ∂ w , x del_(w,x)\partial_{w, x}∂w,x. This is correct if O x ¯ , w O x ¯ , w O_( bar(x),w)\mathcal{O}_{\bar{x}, w}Ox¯,w is a dvr, which is the case outside of finitely many points w x ¯ X ( m + 1 ) w ∈ x ¯ ∩ X ( m + 1 ) w in bar(x)nnX^((m+1))w \in \bar{x} \cap X^{(m+1)}w∈x¯∩X(m+1); in general, one needs to use a push-forward map in Milnor-Witt K K KKK-theory for finite field extensions to define w , x ∂ w , x del_(w,x)\partial_{w, x}∂w,x.
The twisted Milnor-Witt sheaf K n M W ( L ) X K n M W ( L ) X K_(n)^(MW)(L)_(X)\mathcal{K}_{n}^{\mathrm{MW}}(\mathscr{L})_{X}KnMW(L)X is the Nisnevich sheaf on X X XXX associated to the presheaf
U H 0 ( RS ( U , L O U , n ) ) U ↦ H 0 RS ∗ ⁡ U , L ⊗ O U , n U|->H^(0)(RS^(**)(U,LoxO_(U),n))U \mapsto H^{0}\left(\operatorname{RS}^{*}\left(U, \mathscr{L} \otimes \mathcal{O}_{U}, n\right)\right)U↦H0(RS∗⁡(U,L⊗OU,n))
The codimension n n nnn twisted Chow-Witt group of X , C H n ~ ( X ; L ) X , C H n ~ ( X ; L ) X, widetilde(CH^(n))(X;L)X, \widetilde{\mathrm{CH}^{n}}(X ; \mathscr{L})X,CHn~(X;L), is defined as
C H ~ n ( X ; L ) := H n ( RS ( X , L , n ) ) C H ~ n ( X ; L ) := H n RS ∗ ⁡ ( X , L , n ) widetilde(CH)^(n)(X;L):=H^(n)(RS^(**)(X,L,n))\widetilde{\mathrm{CH}}^{n}(X ; \mathscr{L}):=H^{n}\left(\operatorname{RS}^{*}(X, \mathscr{L}, n)\right)CH~n(X;L):=Hn(RS∗⁡(X,L,n))
For details, see [93, CHAP. 5] or [31, cHAP. 2].
For Milnor K K KKK-theory, one has the Gersten complex G ( X , n ) G ∗ ( X , n ) G^(**)(X,n)G^{*}(X, n)G∗(X,n),
G ( X , n ) := x X ( 0 ) K n M W ( k ( x ) ) 0 n m + 1 x X ( m ) K n m M ( k ( x ) ) n m n 1 x X ( n ) K 0 M ( k ( x ) ) , G ∗ ( X , n ) := ⨁ x ∈ X ( 0 )   K n M W ( k ( x ) ) → ∂ 0 ⋯ → ∂ n − m + 1 ⨁ x ∈ X ( m )   K n − m M ( k ( x ) ) → ∂ n − m ⋯ → ∂ n − 1 ⨁ x ∈ X ( n )   K 0 M ( k ( x ) ) , {:[G^(**)(X","n):=bigoplus_(x inX^((0)))K_(n)^(MW)(k(x))rarr"del^(0)"cdotsrarr"del^(n-m+1)"bigoplus_(x inX^((m)))K_(n-m)^(M)(k(x))],[rarr"del^(n-m)"cdotsrarr"del^(n-1)"bigoplus_(x inX^((n)))K_(0)^(M)(k(x))","]:}\begin{aligned} & G^{*}(X, n):=\bigoplus_{x \in X^{(0)}} K_{n}^{\mathrm{MW}}(k(x)) \xrightarrow{\partial^{0}} \cdots \xrightarrow{\partial^{n-m+1}} \bigoplus_{x \in X^{(m)}} K_{n-m}^{M}(k(x)) \\ & \xrightarrow{\partial^{n-m}} \cdots \xrightarrow{\partial^{n-1}} \bigoplus_{x \in X^{(n)}} K_{0}^{M}(k(x)), \end{aligned}G∗(X,n):=⨁x∈X(0)KnMW(k(x))→∂0⋯→∂n−m+1⨁x∈X(m)Kn−mM(k(x))→∂n−m⋯→∂n−1⨁x∈X(n)K0M(k(x)),
with essentially the same definition as the Rost-Schmid complex, without the twisting. This gives us the Milnor K K KKK-theory sheaf K n , X M := ker 0 K n , X M := ker ⁡ ∂ 0 K_(n,X)^(M):=ker del^(0)\mathcal{K}_{n, X}^{M}:=\operatorname{ker} \partial^{0}Kn,XM:=ker⁡∂0, and it follows easily from the definitions that C H n ( X ) = H n ( G ( X , n ) ) C H n ( X ) = H n G ∗ ( X , n ) CH^(n)(X)=H^(n)(G^(**)(X,n))\mathrm{CH}^{n}(X)=H^{n}\left(G^{*}(X, n)\right)CHn(X)=Hn(G∗(X,n)). The same ideas that lead to the Bloch-Kato formula [78]
C H n ( X ) H n ( X N i s , K n , X M ) C H n ( X ) ≅ H n X N i s , K n , X M CH^(n)(X)~=H^(n)(X_(Nis),K_(n,X)^(M))\mathrm{CH}^{n}(X) \cong H^{n}\left(X_{\mathrm{Nis}}, \mathcal{K}_{n, X}^{M}\right)CHn(X)≅Hn(XNis,Kn,XM)
give the isomorphism
C H ~ n ( X ; L ) H n ( X N i s , K n M W ( L ) X ) C H ~ n ( X ; L ) ≅ H n X N i s , K n M W ( L ) X widetilde(CH)^(n)(X;L)~=H^(n)(X_(Nis),K_(n)^(MW)(L)_(X))\widetilde{\mathrm{CH}}^{n}(X ; \mathscr{L}) \cong H^{n}\left(X_{\mathrm{Nis}}, \mathcal{K}_{n}^{\mathrm{MW}}(\mathscr{L})_{X}\right)CH~n(X;L)≅Hn(XNis,KnMW(L)X)
(see the discussion following [31, DEFINITION 3.1] for details). The maps K n M W K n M K n M W → K n M K_(n)^(MW)rarrK_(n)^(M)\mathcal{K}_{n}^{\mathrm{MW}} \rightarrow \mathcal{K}_{n}^{\mathrm{M}}KnMW→KnM give the map of complexes RS ( X , L , n ) G ( X , n ) RS ∗ ⁡ ( X , L , n ) → G ∗ ( X , n ) RS^(**)(X,L,n)rarrG^(**)(X,n)\operatorname{RS}^{*}(X, \mathscr{L}, n) \rightarrow G^{*}(X, n)RS∗⁡(X,L,n)→G∗(X,n) and the corresponding map rnk X , n X , n _(X,n){ }_{X, n}X,n : C H ~ n ( X ; L ) C H n ( X ) C H ~ n ( X ; L ) → C H n ( X ) widetilde(CH)^(n)(X;L)rarrCH^(n)(X)\widetilde{\mathrm{CH}}^{n}(X ; \mathscr{L}) \rightarrow \mathrm{CH}^{n}(X)CH~n(X;L)→CHn(X).
The twists by an invertible sheaf are not just a device for defining the Rost-Schmid complexes and the Chow-Witt groups, they play an integral part in the structure of the overall theory. The Chow groups of smooth varieties admit the functorialities of a Borel-Moore homology theory: they have functorial pullback maps f : C H n ( Y ) C H n ( X ) f ∗ : C H n ( Y ) → C H n ( X ) f^(**):CH^(n)(Y)rarrCH^(n)(X)f^{*}: \mathrm{CH}^{n}(Y) \rightarrow \mathrm{CH}^{n}(X)f∗:CHn(Y)→CHn(X) for each morphism f : X Y f : X → Y f:X rarr Yf: X \rightarrow Yf:X→Y in Sm k Sm k Sm_(k)\operatorname{Sm}_{k}Smk, and for f : X Y f : X → Y f:X rarr Yf: X \rightarrow Yf:X→Y a proper morphism of relative dimension d d ddd, one has the functorial proper push-forward map f : C H n ( X ) C H n d ( Y ) f ∗ : C H n ( X ) → C H n − d ( Y ) f_(**):CH^(n)(X)rarrCH^(n-d)(Y)f_{*}: \mathrm{CH}^{n}(X) \rightarrow \mathrm{CH}^{n-d}(Y)f∗:CHn(X)→CHn−d(Y). The ChowWitt groups also have a contravariant functoriality; for f : X Y f : X → Y f:X rarr Yf: X \rightarrow Yf:X→Y, and L L L\mathscr{L}L an invertible sheaf on Y Y YYY, one has the functorial pullback
f : C H ~ n ( Y , L ) C H ~ n ( X , f L ) f ∗ : C H ~ n ( Y , L ) → C H ~ n X , f ∗ L f^(**): widetilde(CH)^(n)(Y,L)rarr widetilde(CH)^(n)(X,f^(**)L)f^{*}: \widetilde{\mathrm{CH}}^{n}(Y, \mathscr{L}) \rightarrow \widetilde{\mathrm{CH}}^{n}\left(X, f^{*} \mathscr{L}\right)f∗:CH~n(Y,L)→CH~n(X,f∗L)
But for the proper push-forward, one needs to include the orientation sheaf, this being the usual relative dualizing sheaf ω f := ω X / k f ω Y / k 1 ω f := ω X / k ⊗ f ∗ ω Y / k − 1 omega_(f):=omega_(X//k)oxf^(**)omega_(Y//k)^(-1)\omega_{f}:=\omega_{X / k} \otimes f^{*} \omega_{Y / k}^{-1}ωf:=ωX/k⊗f∗ωY/k−1, where ω X / k := det Ω X / k 1 ω X / k := det ⁡ Ω X / k 1 omega_(X//k):=det Omega_(X//k)^(1)\omega_{X / k}:=\operatorname{det} \Omega_{X / k}^{1}ωX/k:=det⁡ΩX/k1 is the sheaf of top-dimensional forms. The push-forward takes the form
f : C H ~ n ( X , ω f f L ) C H ~ n d ( Y , L ) f ∗ : C H ~ n X , ω f ⊗ f ∗ L → C H ~ n − d ( Y , L ) f_(**): widetilde(CH)^(n)(X,omega_(f)oxf^(**)L)rarr widetilde(CH)^(n-d)(Y,L)f_{*}: \widetilde{\mathrm{CH}}^{n}\left(X, \omega_{f} \otimes f^{*} \mathscr{L}\right) \rightarrow \widetilde{\mathrm{CH}}^{n-d}(Y, \mathscr{L})f∗:CH~n(X,ωf⊗f∗L)→CH~n−d(Y,L)
This limits the possible twists C H ~ n ( X , M ) C H ~ n ( X , M ) widetilde(CH)^(n)(X,M)\widetilde{\mathrm{CH}}^{n}(X, \mathcal{M})CH~n(X,M) for which a push-forward f f ∗ f_(**)f_{*}f∗ is even defined; this type of restricted push-forward is typical of so-called SL-oriented theories, such as hermitian K K KKK-theory. See [1] for a detailed discussion of SL-oriented theories and [31, chAP. 3] for the details concerning the push-forward in C H ~ C H ~ ∗ widetilde(CH)^(**)\widetilde{\mathrm{CH}}^{*}CH~∗.

4.2. The homotopy t t ttt-structure and Morel's theorem

Building on the Bloch-Kato formula, C H n ( X ) H n ( X N i s , K n , X M ) C H n ( X ) ≅ H n X N i s , K n , X M CH^(n)(X)~=H^(n)(X_(Nis),K_(n,X)^(M))\mathrm{CH}^{n}(X) \cong H^{n}\left(X_{\mathrm{Nis}}, \mathcal{K}_{n, X}^{M}\right)CHn(X)≅Hn(XNis,Kn,XM), one can construct a good bigraded cohomology theory EM ( K M ) EM ⁡ K ∗ M ∗ ∗ EM (K_(**)^(M))^(****)\operatorname{EM}\left(\mathcal{K}_{*}^{M}\right)^{* *}EM⁡(K∗M)∗∗ by using all the cohomology groups. To get the correct bigrading, one should set
EM ( K M ) a , b ( X ) := H a b ( X N i s , K b M ) EM ⁡ K ∗ M a , b ( X ) := H a − b X N i s , K b M EM (K_(**)^(M))^(a,b)(X):=H^(a-b)(X_(Nis),K_(b)^(M))\operatorname{EM}\left(\mathcal{K}_{*}^{M}\right)^{a, b}(X):=H^{a-b}\left(X_{\mathrm{Nis}}, \mathcal{K}_{b}^{M}\right)EM⁡(K∗M)a,b(X):=Ha−b(XNis,KbM)
giving in particular EM ( K M ) 2 n , n ( X ) = C H n ( X ) EM ⁡ K ∗ M 2 n , n ( X ) = C H n ( X ) EM (K_(**)^(M))^(2n,n)(X)=CH^(n)(X)\operatorname{EM}\left(\mathcal{K}_{*}^{M}\right)^{2 n, n}(X)=\mathrm{CH}^{n}(X)EM⁡(K∗M)2n,n(X)=CHn(X). It was recognized early on that this theory is not the sought-after motivic cohomology, for instance, for X = Spec F , F X = Spec ⁡ F , F X=Spec F,FX=\operatorname{Spec} F, FX=Spec⁡F,F a field, one gets exactly the Milnor K K KKK-theory of F F FFF, and none of the other parts of the K K KKK-theory of F F FFF. In spite of this, this theory and the similarly defined theory for Milnor-Witt K K KKK-theory have a natural place in the universe of motivic cohomology theories, which we now explain.
The classical stable homotopy category S H S H SH\mathrm{SH}SH is a triangulated category with a natural t t ttt-structure measuring connectedness, mentioned in Section 2.5. For S H S H SH\mathrm{SH}SH, the truncations give the terms in the Moore-Postnikov tower
τ n + 1 E τ n E E ⋯ → Ï„ ≥ n + 1 E → Ï„ ≥ n E → ⋯ → E cdots rarrtau_( >= n+1)E rarrtau_( >= n)E rarr cdots rarr E\cdots \rightarrow \tau_{\geq n+1} E \rightarrow \tau_{\geq n} E \rightarrow \cdots \rightarrow E⋯→τ≥n+1E→τ≥nE→⋯→E
with τ n E E Ï„ ≥ n E → E tau_( >= n)E rarr E\tau_{\geq n} E \rightarrow Eτ≥nE→E characterized by inducing an isomorphism on π m Ï€ m pi_(m)\pi_{m}Ï€m for m n m ≥ n m >= nm \geq nm≥n and with π m τ n E = 0 Ï€ m Ï„ ≥ n E = 0 pi_(m)tau_( >= n)E=0\pi_{m} \tau_{\geq n} E=0Ï€mτ≥nE=0 for m < n m < n m < nm<nm<n. The heart of SH is the category of spectra E E EEE with π m E = 0 Ï€ m E = 0 pi_(m)E=0\pi_{m} E=0Ï€mE=0 for m 0 m ≠ 0 m!=0m \neq 0m≠0, which are just the Eilenberg-MacLane spectra EM ( A ) , A EM ⁡ ( A ) , A EM(A),A\operatorname{EM}(A), AEM⁡(A),A an abelian group. Thus, the heart of S H S H SH\mathrm{SH}SH is A b A b Ab\mathbf{A b}Ab and the cohomology theory represented by τ 0 E Ï„ 0 E tau_(0)E\tau_{0} EÏ„0E is
EM ( π 0 E ) n ( X ) := H n ( X , π 0 E ) EM ⁡ Ï€ 0 E n ( X ) := H n X , Ï€ 0 E EM (pi_(0)E)^(n)(X):=H^(n)(X,pi_(0)E)\operatorname{EM}\left(\pi_{0} E\right)^{n}(X):=H^{n}\left(X, \pi_{0} E\right)EM⁡(Ï€0E)n(X):=Hn(X,Ï€0E)
singular cohomology with coefficients in the abelian group π 0 E Ï€ 0 E pi_(0)E\pi_{0} EÏ€0E.
We have a parallel t t ttt-structure on S H ( k ) S H ( k ) SH(k)\mathrm{SH}(k)SH(k), introduced by Morel [92, §5.2], called the homotopy t t ttt-structure (and not coming from Voevodsky's slice tower discussed in Section 2.5). This is similar to the t t ttt-structure on S H S H SH\mathrm{SH}SH, where one takes into account the fact that one has bigraded homotopy sheaves π a , b E Ï€ a , b E pi_(a,b)E\pi_{a, b} EÏ€a,bE for E S H ( k ) E ∈ S H ( k ) E inSH(k)E \in \mathrm{SH}(k)E∈SH(k), rather than a Z Z Z\mathbb{Z}Z-graded family of homotopy groups π n E Ï€ n E pi_(n)E\pi_{n} EÏ€nE for E S H E ∈ S H E inSHE \in \mathrm{SH}E∈SH. The truncation τ n E Ï„ ≥ n E tau_( >= n)E\tau_{\geq n} Eτ≥nE is characterized by
π a , b ( τ n E ) = { π a , b ( E ) if a b n 0 if a b < n Ï€ a , b Ï„ ≥ n E = Ï€ a , b ( E )  if  a − b ≥ n 0  if  a − b < n pi_(a,b)(tau_( >= n)E)={[pi_(a,b)(E)," if "a-b >= n],[0," if "a-b < n]:}\pi_{a, b}\left(\tau_{\geq n} E\right)= \begin{cases}\pi_{a, b}(E) & \text { if } a-b \geq n \\ 0 & \text { if } a-b<n\end{cases}Ï€a,b(τ≥nE)={Ï€a,b(E) if a−b≥n0 if a−b<n
Recalling that the sphere S a , b S a , b S^(a,b)S^{a, b}Sa,b is S a b G m b S a − b ∧ G m b S^(a-b)^^G_(m)^(b)S^{a-b} \wedge \mathbb{G}_{m}^{b}Sa−b∧Gmb, the homotopy t t ttt-structure on S H ( k ) S H ( k ) SH(k)\mathrm{SH}(k)SH(k) is measuring S 1 S 1 S^(1)S^{1}S1-connectedness, instead of the P 1 P 1 P^(1)\mathbb{P}^{1}P1-connectedness measured by Voevodsky's slice tower.
We denote the 0 th truncation τ 0 E Ï„ 0 E tau_(0)E\tau_{0} EÏ„0E for E S H ( k ) E ∈ S H ( k ) E inSH(k)E \in \mathrm{SH}(k)E∈SH(k) by EM ( π , E ) EM ⁡ Ï€ − ∗ , − ∗ E EM(pi_(-**,-**)E)\operatorname{EM}\left(\pi_{-*,-*} E\right)EM⁡(π−∗,−∗E); the notation comes from Morel's identification of the heart with his category of homotopy modules; for details, see [92, §5.2]. The corresponding cohomology theory satisfies, for X S m k X ∈ S m k X inSm_(k)X \in \mathrm{Sm}_{k}X∈Smk,
EM ( π , E ) a , b ( X ) = H a b ( X N i s , π b , b ( E ) ) EM ⁡ Ï€ − ∗ , − ∗ E a , b ( X ) = H a − b X N i s , Ï€ − b , − b ( E ) EM (pi_(-**,-**)E)^(a,b)(X)=H^(a-b)(X_(Nis),pi_(-b,-b)(E))\operatorname{EM}\left(\pi_{-*,-*} E\right)^{a, b}(X)=H^{a-b}\left(X_{\mathrm{Nis}}, \pi_{-b,-b}(E)\right)EM⁡(π−∗,−∗E)a,b(X)=Ha−b(XNis,π−b,−b(E))
Here we have Morel's fundamental theorem [92, THEOREM 6.4.1] computing τ 0 Ï„ 0 tau_(0)\tau_{0}Ï„0 of the sphere spectrum 1 k S H ( k ) 1 k ∈ S H ( k ) 1_(k)inSH(k)1_{k} \in \mathrm{SH}(k)1k∈SH(k).
Theorem 4.4 (Morel). Let k k kkk be a perfect field. Then there are canonical isomorphisms of sheaves on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk
π n , n ( 1 k ) = K n M W Ï€ − n , − n 1 k = K n M W pi_(-n,-n)(1_(k))=K_(n)^(MW)\pi_{-n,-n}\left(1_{k}\right)=\mathcal{K}_{n}^{\mathrm{MW}}π−n,−n(1k)=KnMW
Consequently,
τ 0 1 k = EM ( K M W ) Ï„ 0 1 k = EM ⁡ K ∗ M W tau_(0)1_(k)=EM(K_(**)^(MW))\tau_{0} 1_{k}=\operatorname{EM}\left(\mathcal{K}_{*}^{\mathrm{MW}}\right)Ï„01k=EM⁡(K∗MW)
and
EM ( K M W ) a , b ( X ) = H a b ( X N i s , K b , X M W ) EM ⁡ K ∗ M W a , b ( X ) = H a − b X N i s , K b , X M W EM (K_(**)^(MW))^(a,b)(X)=H^(a-b)(X_(Nis),K_(b,X)^(MW))\operatorname{EM}\left(\mathcal{K}_{*}^{\mathrm{MW}}\right)^{a, b}(X)=H^{a-b}\left(X_{\mathrm{Nis}}, \mathcal{K}_{b, X}^{\mathrm{MW}}\right)EM⁡(K∗MW)a,b(X)=Ha−b(XNis,Kb,XMW)
Going back in time a bit, we have the theorem of Totaro [115] and Nesterenko-Suslin [96]
H n ( F , Z ( n ) ) K n M ( F ) H n ( F , Z ( n ) ) ≅ K n M ( F ) H^(n)(F,Z(n))~=K_(n)^(M)(F)H^{n}(F, \mathbb{Z}(n)) \cong K_{n}^{M}(F)Hn(F,Z(n))≅KnM(F)
for F F FFF a field. Combined with the isomorphism
s 0 1 k H Z s 0 1 k ≅ H Z s_(0)1_(k)~=HZs_{0} 1_{k} \cong H \mathbb{Z}s01k≅HZ
of [ 9 , 85 , 122 ] [ 9 , 85 , 122 ] [9,85,122][9,85,122][9,85,122], we have
Theorem 4.5. Let k k kkk be a perfect field. Then
τ 0 s 0 1 k = τ 0 H Z = EM ( K M ) Ï„ 0 s 0 1 k = Ï„ 0 H Z = EM ⁡ K ∗ M tau_(0)s_(0)1_(k)=tau_(0)HZ=EM(K_(**)^(M))\tau_{0} s_{0} 1_{k}=\tau_{0} H \mathbb{Z}=\operatorname{EM}\left(\mathcal{K}_{*}^{M}\right)Ï„0s01k=Ï„0HZ=EM⁡(K∗M)
and
EM ( K M ) a , b ( X ) = H a b ( X N i s , K b , X M ) EM ⁡ K ∗ M a , b ( X ) = H a − b X N i s , K b , X M EM (K_(**)^(M))^(a,b)(X)=H^(a-b)(X_(Nis),K_(b,X)^(M))\operatorname{EM}\left(\mathcal{K}_{*}^{M}\right)^{a, b}(X)=H^{a-b}\left(X_{\mathrm{Nis}}, \mathcal{K}_{b, X}^{M}\right)EM⁡(K∗M)a,b(X)=Ha−b(XNis,Kb,XM)
for X Sm k X ∈ Sm k X inSm_(k)X \in \operatorname{Sm}_{k}X∈Smk.
Bachmann proves an extension of this result. Recall Voevodsky's slice tower
f n + 1 E f n E f 0 E E ⋯ → f n + 1 E → f n E → ⋯ → f 0 E → ⋯ → E cdots rarrf_(n+1)E rarrf_(n)E rarr cdots rarrf_(0)E rarr cdots rarr E\cdots \rightarrow f_{n+1} E \rightarrow f_{n} E \rightarrow \cdots \rightarrow f_{0} E \rightarrow \cdots \rightarrow E⋯→fn+1E→fnE→⋯→f0E→⋯→E
with s n E s n E s_(n)Es_{n} EsnE the layer given by the distinguished triangle
f n + 1 E f n E s n E f n + 1 E [ 1 ] f n + 1 E → f n E → s n E → f n + 1 E [ 1 ] f_(n+1)E rarrf_(n)E rarrs_(n)E rarrf_(n+1)E[1]f_{n+1} E \rightarrow f_{n} E \rightarrow s_{n} E \rightarrow f_{n+1} E[1]fn+1E→fnE→snE→fn+1E[1]
Recall that this is not the truncation tower of a t t ttt-structure, as the subcategories defined by the layers s n := f n / f n + 1 s n := f n / f n + 1 s_(n):=f_(n)//f_(n+1)s_{n}:=f_{n} / f_{n+1}sn:=fn/fn+1 are triangulated categories, not abelian categories.
Proposition 4.6 ([7, LEMMA 12]). Let 1 k EM ( K M ) 1 k → EM ⁡ K ∗ M 1_(k)rarr EM(K_(**)^(M))1_{k} \rightarrow \operatorname{EM}\left(\mathcal{K}_{*}^{M}\right)1k→EM⁡(K∗M) be the composition 1 k τ 0 1 k = 1 k → Ï„ 0 1 k = 1_(k)rarrtau_(0)1_(k)=1_{k} \rightarrow \tau_{0} 1_{k}=1k→τ01k= EM ( K M W ) EM ( K M ) EM ⁡ K ∗ M W → EM ⁡ K ∗ M EM(K_(**)^(MW))rarr EM(K_(**)^(M))\operatorname{EM}\left(\mathcal{K}_{*}^{\mathrm{MW}}\right) \rightarrow \operatorname{EM}\left(\mathcal{K}_{*}^{M}\right)EM⁡(K∗MW)→EM⁡(K∗M), the latter map induced by the surjection K M W K M K ∗ M W → K ∗ M K_(**)^(MW)rarrK_(**)^(M)\mathcal{K}_{*}^{\mathrm{MW}} \rightarrow \mathcal{K}_{*}^{M}K∗MW→K∗M. Then the induced maps
s 0 ( 1 k ) s 0 EM ( K M ) f 0 EM ( K M ) = f 0 τ 0 H Z s 0 1 k → s 0 EM ⁡ K ∗ M ← f 0 EM ⁡ K ∗ M = f 0 Ï„ 0 H Z s_(0)(1_(k))rarrs_(0)EM(K_(**)^(M))larrf_(0)EM(K_(**)^(M))=f_(0)tau_(0)HZs_{0}\left(1_{k}\right) \rightarrow s_{0} \operatorname{EM}\left(\mathcal{K}_{*}^{M}\right) \leftarrow f_{0} \operatorname{EM}\left(\mathcal{K}_{*}^{M}\right)=f_{0} \tau_{0} H \mathbb{Z}s0(1k)→s0EM⁡(K∗M)←f0EM⁡(K∗M)=f0Ï„0HZ
are all isomorphisms, so all of these objects are isomorphic to the motivic cohomology spectrum H Z H Z HZH \mathbb{Z}HZ.
The truncation functors for the homotopy t t ttt-structure and for the Voevodsky slice tower do not commute. Since 1 k 1 k 1_(k)1_{k}1k is effective, we have f 0 1 k = 1 k f 0 1 k = 1 k f_(0)1_(k)=1_(k)f_{0} 1_{k}=1_{k}f01k=1k and so τ 0 f 0 1 k = τ 0 1 k = Ï„ 0 f 0 1 k = Ï„ 0 1 k = tau_(0)f_(0)1_(k)=tau_(0)1_(k)=\tau_{0} f_{0} 1_{k}=\tau_{0} 1_{k}=Ï„0f01k=Ï„01k= EM ( K M W ) EM ⁡ K ∗ M W EM(K_(**)^(MW))\operatorname{EM}\left(\mathcal{K}_{*}^{\mathrm{MW}}\right)EM⁡(K∗MW). The truncations in the other order give us something new.

4.3. Milnor-Witt motivic cohomology

Definition 4.7 ([7, notation, P. 1134, JUSt BEFORE LEMMA 12]). Let k k kkk be a perfect field. Define the Milnor-Witt motivic cohomology spectrum H ~ Z S H ( k ) eff H ~ Z ∈ S H ( k ) eff  tilde(H)ZinSH(k)^("eff ")\tilde{H} \mathbb{Z} \in \mathrm{SH}(k)^{\text {eff }}H~Z∈SH(k)eff  by
H ~ Z := f 0 ( τ 0 1 k ) = f 0 E M ( K M W ) H ~ Z := f 0 Ï„ 0 1 k = f 0 E M K ∗ M W tilde(H)Z:=f_(0)(tau_(0)1_(k))=f_(0)EM(K_(**)^(MW))\tilde{H} \mathbb{Z}:=f_{0}\left(\tau_{0} 1_{k}\right)=f_{0} \mathrm{EM}\left(\mathcal{K}_{*}^{\mathrm{MW}}\right)H~Z:=f0(Ï„01k)=f0EM(K∗MW)
The canonical map τ 0 1 k τ 0 s 0 1 k = τ 0 H Z Ï„ 0 1 k → Ï„ 0 s 0 1 k = Ï„ 0 H Z tau_(0)1_(k)rarrtau_(0)s_(0)1_(k)=tau_(0)HZ\tau_{0} 1_{k} \rightarrow \tau_{0} s_{0} 1_{k}=\tau_{0} H \mathbb{Z}Ï„01k→τ0s01k=Ï„0HZ induces the map
H ~ Z = f 0 ( τ 0 1 k ) Ξ f 0 τ 0 H Z = H Z . H ~ Z = f 0 Ï„ 0 1 k → Ξ f 0 Ï„ 0 H Z = H Z . tilde(H)Z=f_(0)(tau_(0)1_(k))rarr"Xi"f_(0)tau_(0)HZ=HZ.\tilde{H} \mathbb{Z}=f_{0}\left(\tau_{0} 1_{k}\right) \xrightarrow{\Xi} f_{0} \tau_{0} H \mathbb{Z}=H \mathbb{Z} .H~Z=f0(Ï„01k)→Ξf0Ï„0HZ=HZ.
For X Sm k X ∈ Sm k X inSm_(k)X \in \operatorname{Sm}_{k}X∈Smk, the Milnor-Witt motivic cohomology in bidegree ( a , b ) ( a , b ) (a,b)(a, b)(a,b) is defined as H ~ Z a , b ( X ) H ~ Z a , b ( X ) tilde(H)Z^(a,b)(X)\tilde{H} \mathbb{Z}^{a, b}(X)H~Za,b(X).
Remarkably, one can compute H ~ Z a , b ( X ) H ~ Z a , b ( X ) tilde(H)Z^(a,b)(X)\tilde{H} \mathbb{Z}^{a, b}(X)H~Za,b(X) in terms of the Milnor-Witt sheaves, at least for some of the indices ( a , b ) ( a , b ) (a,b)(a, b)(a,b); one also recovers the Chow-Witt groups. For X = Spec F X = Spec ⁡ F X=Spec FX=\operatorname{Spec} FX=Spec⁡F, the spectrum of a field F F FFF, one has a complete computation in terms of the Milnor-Witt K K KKK-groups and the usual motivic cohomology H Z a , b ( X ) := H a ( X , Z ( b ) ) H Z a , b ( X ) := H a ( X , Z ( b ) ) HZ^(a,b)(X):=H^(a)(X,Z(b))H \mathbb{Z}^{a, b}(X):=H^{a}(X, \mathbb{Z}(b))HZa,b(X):=Ha(X,Z(b))
Theorem 4.8 (Bachmann). For X Sm k X ∈ Sm k X inSm_(k)X \in \operatorname{Sm}_{k}X∈Smk and b 0 b ≤ 0 b <= 0b \leq 0b≤0, there are natural isomorphisms
H ~ Z a , b ( X ) H a b ( X N i s , K b , X M W ) = { H a b ( X N i s , W X ) for b < 0 H a b ( X N i s , E W X ) for b = 0 H ~ Z a , b ( X ) ≅ H a − b X N i s , K b , X M W = H a − b X N i s , W X  for  b < 0 H a − b X N i s , E W X  for  b = 0 tilde(H)Z^(a,b)(X)~=H^(a-b)(X_(Nis),K_(b,X)^(MW))={[H^(a-b)(X_(Nis),W_(X))," for "b < 0],[H^(a-b)(X_(Nis),EW_(X))," for "b=0]:}\tilde{H} \mathbb{Z}^{a, b}(X) \cong H^{a-b}\left(X_{\mathrm{Nis}}, \mathcal{K}_{b, X}^{\mathrm{MW}}\right)= \begin{cases}H^{a-b}\left(X_{\mathrm{Nis}}, \mathcal{W}_{X}\right) & \text { for } b<0 \\ H^{a-b}\left(X_{\mathrm{Nis}}, \mathcal{E} \mathcal{W}_{X}\right) & \text { for } b=0\end{cases}H~Za,b(X)≅Ha−b(XNis,Kb,XMW)={Ha−b(XNis,WX) for b<0Ha−b(XNis,EWX) for b=0
Here W X W X W_(X)\mathcal{W}_{X}WX is the sheaf of Witt groups and E W X E W X EW_(X)\mathscr{E} \mathcal{W}_{X}EWX is the sheaf of Grothendieck-Witt rings.
For X S m k X ∈ S m k X inSm_(k)X \in \mathrm{Sm}_{k}X∈Smk and n Z n ∈ Z n inZn \in \mathbb{Z}n∈Z, we have
H ~ Z 2 n , n ( X ) C H ~ n ( X ) H ~ Z 2 n , n ( X ) ≅ C H ~ n ( X ) tilde(H)Z^(2n,n)(X)~= widetilde(CH)^(n)(X)\tilde{H} \mathbb{Z}^{2 n, n}(X) \cong \widetilde{C H}^{n}(X)H~Z2n,n(X)≅CH~n(X)
For F F FFF a field, we have isomorphisms
H ~ Z a , b ( Spec F ) { K n M W ( F ) for a = b = n H Z a , b ( Spec F ) for a b H ~ Z a , b ( Spec ⁡ F ) ≅ K n M W ( F )  for  a = b = n H Z a , b ( Spec ⁡ F )  for  a ≠ b tilde(H)Z^(a,b)(Spec F)~={[K_(n)^(MW)(F)," for "a=b=n],[HZ^(a,b)(Spec F)," for "a!=b]:}\tilde{H} \mathbb{Z}^{a, b}(\operatorname{Spec} F) \cong \begin{cases}K_{n}^{\mathrm{MW}}(F) & \text { for } a=b=n \\ H \mathbb{Z}^{a, b}(\operatorname{Spec} F) & \text { for } a \neq b\end{cases}H~Za,b(Spec⁡F)≅{KnMW(F) for a=b=nHZa,b(Spec⁡F) for a≠b
This follows from
Theorem 4.9 ([7, THEOREM 17]). Let H ~ Z a , b , H Z a , b H ~ Z a , b , H Z a , b tilde(H)Z^(a,b),HZ^(a,b)\tilde{\mathscr{H}} \mathbb{Z}^{a, b}, \mathscr{H} \mathbb{Z}^{a, b}H~Za,b,HZa,b denote the respective homotopy sheaves π a , b ( H ~ Z ) , π a , b ( H Z ) Ï€ − a , − b ( H ~ Z ) , Ï€ − a , − b ( H Z ) pi_(-a,-b)( tilde(H)Z),pi_(-a,-b)(HZ)\pi_{-a,-b}(\tilde{H} \mathbb{Z}), \pi_{-a,-b}(H \mathbb{Z})π−a,−b(H~Z),π−a,−b(HZ). Then for a b a ≠ b a!=ba \neq ba≠b, the map
Ξ a , b : H ~ Z a , b H Z a , b Ξ a , b : H ~ Z a , b → H Z a , b Xi^(a,b): tilde(H)Z^(a,b)rarrHZ^(a,b)\Xi^{a, b}: \tilde{\mathscr{H}} \mathbb{Z}^{a, b} \rightarrow \mathscr{H} \mathbb{Z}^{a, b}Ξa,b:H~Za,b→HZa,b
is an isomorphism. Moreover, we have canonical isomorphisms H ~ Z b , b = K b M W H ~ Z b , b = K b M W tilde(H)Z^(b,b)=K_(b)^(MW)\tilde{\mathscr{H}} \mathbb{Z}^{b, b}=\mathcal{K}_{b}^{\mathrm{MW}}H~Zb,b=KbMW, H Z b , b = K b M H Z b , b = K b M HZ^(b,b)=K_(b)^(M)\mathscr{H} \mathbb{Z}^{b, b}=\mathcal{K}_{b}^{M}HZb,b=KbM, and Ξ a , b : H ~ Z b , b H b , b Ξ a , b : H ~ Z b , b → H b , b Xi^(a,b): tilde(H)Z^(b,b)rarrH^(b,b)\Xi^{a, b}: \tilde{\mathscr{H}} \mathbb{Z}^{b, b} \rightarrow \mathscr{H}^{b, b}Ξa,b:H~Zb,b→Hb,b is canonical surjection K b M W K b M K b M W → K b M K_(b)^(MW)rarrK_(b)^(M)\mathcal{K}_{b}^{\mathrm{MW}} \rightarrow \mathcal{K}_{b}^{M}KbMW→KbM.
To prove Theorem 4.8, one applies this to the local-global spectral sequence
E 2 p , q ( n ) := H p ( X N i s , H ~ Z q , n ) H ~ Z p + q , n ( X ) E 2 p , q ( n ) := H p X N i s , H ~ Z q , n ⇒ H ~ Z p + q , n ( X ) E_(2)^(p,q)(n):=H^(p)(X_(Nis),( tilde(H))Z^(q,n))=> tilde(H)Z^(p+q,n)(X)E_{2}^{p, q}(n):=H^{p}\left(X_{\mathrm{Nis}}, \tilde{\mathscr{H}} \mathbb{Z}^{q, n}\right) \Rightarrow \tilde{H} \mathbb{Z}^{p+q, n}(X)E2p,q(n):=Hp(XNis,H~Zq,n)⇒H~Zp+q,n(X)
noting that H Z q , n = 0 H Z q , n = 0 HZ^(q,n)=0\mathscr{H} \mathbb{Z}^{q, n}=0HZq,n=0 for n < 0 n < 0 n < 0n<0n<0. This implies that the Gersten resolution of H Z q , n H Z q , n HZ^(q,n)\mathscr{H} \mathbb{Z}^{q, n}HZq,n has length n ≤ n <= n\leq n≤n and thus H p ( X Nis , H Z q , n ) = 0 H p X Nis  , H Z q , n = 0 H^(p)(X_("Nis "),HZ^(q,n))=0H^{p}\left(X_{\text {Nis }}, \mathscr{H} \mathbb{Z}^{q, n}\right)=0Hp(XNis ,HZq,n)=0 for p > n p > n p > np>np>n.
In general, one can approximate H ~ Z a , b ( X ) H ~ Z a , b ( X ) tilde(H)Z^(a,b)(X)\tilde{H} \mathbb{Z}^{a, b}(X)H~Za,b(X) using the local-global sequence. Combined with Theorem 4.9 and the exact sheaf sequence
0 n + 1 K n M W K n M 0 0 → ℓ n + 1 → K n M W → K n M → 0 0rarrℓ^(n+1)rarrK_(n)^(MW)rarrK_(n)^(M)rarr00 \rightarrow \ell^{n+1} \rightarrow \mathcal{K}_{n}^{\mathrm{MW}} \rightarrow \mathcal{K}_{n}^{M} \rightarrow 00→ℓn+1→KnMW→KnM→0
this tells us that the Milnor-Witt cohomology of X X XXX is built out of the usual motivic cohomology combined with information arising from quadratic forms.

4.4. Milnor-Witt motives

Rather than pulling the Milnor-Witt cohomology out of the motivic stable homotopy hat, there is another construction that is embedded in a Voevodsky-type triangulated category built out of a modified category of correspondences. We refer to [8] and [31] for details.
The Chow-Witt groups on a smooth X X XXX have been defined using the Rost-Schmid complex; one can also define Chow-Witt cycles with a fixed support using a modified version of the Rost-Schmid complex.
Definition 4.10. Let X X XXX be a smooth k k kkk-scheme, L L L\mathscr{L}L an invertible sheaf on X X XXX, and T X T ⊂ X T sub XT \subset XT⊂X a closed subset. The n n nnnth L L L\mathscr{L}L-twisted Rost-Schmid complex with supports in T , RS T ( X , n ; L ) T , RS T ∗ ⁡ ( X , n ; L ) T,RS_(T)^(**)(X,n;L)T, \operatorname{RS}_{T}^{*}(X, n ; \mathscr{L})T,RST∗⁡(X,n;L), is the subcomplex of RS ( X , L , n ) RS ∗ ⁡ ( X , L , n ) RS^(**)(X,L,n)\operatorname{RS}^{*}(X, \mathscr{L}, n)RS∗⁡(X,L,n) with
RS T m ( X , L , n ) := x T X ( p ) K n m M W ( k ( x ) ; L x O X , x m ( m x / m x 2 ) ) RS m ( X , L , n ) RS T m ⁡ ( X , L , n ) := ⨁ x ∈ T ∩ X ( p )   K n − m M W k ( x ) ; L x ⊗ O X , x ⋀ m   m x / m x 2 ∨ ⊂ RS m ⁡ ( X , L , n ) RS_(T)^(m)(X,L,n):=bigoplus_(x in T nnX^((p)))K_(n-m)^(MW)(k(x);L_(x)ox_(O_(X,x))^^^m(m_(x)//m_(x)^(2))^(vv))subRS^(m)(X,L,n)\operatorname{RS}_{T}^{m}(X, \mathscr{L}, n):=\bigoplus_{x \in T \cap X^{(p)}} K_{n-m}^{\mathrm{MW}}\left(k(x) ; \mathscr{L}_{x} \otimes_{\mathcal{O}_{X, x}} \bigwedge^{m}\left(\mathfrak{m}_{x} / \mathfrak{m}_{x}^{2}\right)^{\vee}\right) \subset \operatorname{RS}^{m}(X, \mathscr{L}, n)RSTm⁡(X,L,n):=⨁x∈T∩X(p)Kn−mMW(k(x);Lx⊗OX,x⋀m(mx/mx2)∨)⊂RSm⁡(X,L,n)
The usual arguments used to prove Gersten's conjecture yield the following result.
Lemma 4.11. Let X X XXX be a smooth k k kkk-scheme, L L L\mathscr{L}L an invertible sheaf on X X XXX, and T X T ⊂ X T sub XT \subset XT⊂X a closed subset. The cohomology with support H T p ( X , K n M W ( L ) X ) H T p X , K n M W ( L ) X H_(T)^(p)(X,K_(n)^(MW)(L)_(X))H_{T}^{p}\left(X, \mathcal{K}_{n}^{\mathrm{MW}}(\mathscr{L})_{X}\right)HTp(X,KnMW(L)X) is computed as
H T p ( X , K n M W ( L ) X ) = H p ( RS T ( X , L , n ) ) H T p X , K n M W ( L ) X = H p RS T ∗ ⁡ ( X , L , n ) H_(T)^(p)(X,K_(n)^(MW)(L)_(X))=H^(p)(RS_(T)^(**)(X,L,n))H_{T}^{p}\left(X, \mathcal{K}_{n}^{\mathrm{MW}}(\mathscr{L})_{X}\right)=H^{p}\left(\operatorname{RS}_{T}^{*}(X, \mathscr{L}, n)\right)HTp(X,KnMW(L)X)=Hp(RST∗⁡(X,L,n))
Suppose T T TTT has pure codimension n n nnn on X X XXX. Then X ( m ) T = X ( m ) ∩ T = ∅ X^((m))nn T=O/X^{(m)} \cap T=\emptysetX(m)∩T=∅ for m < n , X ( n ) T m < n , X ( n ) ∩ T m < n,X^((n))nn Tm<n, X^{(n)} \cap Tm<n,X(n)∩T is the finite set of generic points T ( 0 ) T ( 0 ) T^((0))T^{(0)}T(0) of T T TTT and X ( n + 1 ) T = T ( 1 ) X ( n + 1 ) ∩ T = T ( 1 ) X^((n+1))nn T=T^((1))X^{(n+1)} \cap T=T^{(1)}X(n+1)∩T=T(1) is the set of codimension one points of T T TTT. This gives us the exact sequence
0 H T n ( X , K n M W ( L ) X ) z T ( 0 ) G W ( k ( z ) , det 1 ( m z / m z 2 ) L ) z T ( 1 ) W ( k ( z ) , det 1 ( m z / m z 2 ) L ) 0 → H T n X , K n M W ( L ) X → ⨁ z ∈ T ( 0 )   G W k ( z ) , det − 1 ⁡ m z / m z 2 ⊗ L → ⨁ z ∈ T ( 1 )   W k ( z ) , det − 1 ⁡ m z / m z 2 ⊗ L {:[0 rarrH_(T)^(n)(X,K_(n)^(MW)(L)_(X))rarrbigoplus_(z inT^((0)))GW(k(z),det^(-1)(m_(z)//m_(z)^(2))oxL)],[ rarrbigoplus_(z inT^((1)))W(k(z),det^(-1)(m_(z)//m_(z)^(2))oxL)]:}\begin{aligned} 0 & \rightarrow H_{T}^{n}\left(X, \mathcal{K}_{n}^{\mathrm{MW}}(\mathscr{L})_{X}\right) \rightarrow \bigoplus_{z \in T^{(0)}} \mathrm{GW}\left(k(z), \operatorname{det}^{-1}\left(\mathfrak{m}_{z} / \mathfrak{m}_{z}^{2}\right) \otimes \mathscr{L}\right) \\ & \rightarrow \bigoplus_{z \in T^{(1)}} W\left(k(z), \operatorname{det}^{-1}\left(\mathfrak{m}_{z} / \mathfrak{m}_{z}^{2}\right) \otimes \mathscr{L}\right) \end{aligned}0→HTn(X,KnMW(L)X)→⨁z∈T(0)GW(k(z),det−1⁡(mz/mz2)⊗L)→⨁z∈T(1)W(k(z),det−1⁡(mz/mz2)⊗L)
which allows us to think of H T n ( X , K n M W ( L ) X ) H T n X , K n M W ( L ) X H_(T)^(n)(X,K_(n)^(MW)(L)_(X))H_{T}^{n}\left(X, \mathcal{K}_{n}^{\mathrm{MW}}(\mathscr{L})_{X}\right)HTn(X,KnMW(L)X) as the group of "Grothendieck-Witt cycles" supported on T T TTT, whose definition we hinted at in the beginning of this section. We write this as Z ~ T n ( X , L , n ) Z ~ T n ( X , L , n ) tilde(Z)_(T)^(n)(X,L,n)\tilde{Z}_{T}^{n}(X, \mathscr{L}, n)Z~Tn(X,L,n), with the warning that this is only defined for T T TTT a closed subset of a smooth X X XXX of pure codimension n n nnn.
Note that the fact that T T TTT has pure codimension n n nnn implies that there are no relations in H T n ( X , K n M W ( L ) X ) H T n X , K n M W ( L ) X H_(T)^(n)(X,K_(n)^(MW)(L)_(X))H_{T}^{n}\left(X, \mathcal{K}_{n}^{\mathrm{MW}}(\mathscr{L})_{X}\right)HTn(X,KnMW(L)X) coming from K 1 M W ( k ( w ) ) K 1 M W ( k ( w ) ) K_(1)^(MW)(k(w))K_{1}^{\mathrm{MW}}(k(w))K1MW(k(w)) for w w www a codimension n 1 n − 1 n-1n-1n−1 point of X X XXX. For similar reasons, the corresponding group for the Chow groups, H T n ( X , K n , X M ) H T n X , K n , X M H_(T)^(n)(X,K_(n,X)^(M))H_{T}^{n}\left(X, \mathcal{K}_{n, X}^{M}\right)HTn(X,Kn,XM), is just the subgroup Z T n ( X ) Z T n ( X ) Z_(T)^(n)(X)Z_{T}^{n}(X)ZTn(X) of Z n ( X ) Z n ( X ) Z^(n)(X)Z^{n}(X)Zn(X) freely generated by the irreducible components of T T TTT, that is, the group of codimension n n nnn cycles on X X XXX with support contained in T T TTT.
For T T X T ⊂ T ′ ⊂ X T subT^(')sub XT \subset T^{\prime} \subset XT⊂T′⊂X, two codimension- n n nnn closed subsets, we have the evident map Z ~ T n ( X , L , n ) Z ~ T n ( X , L , n ) Z ~ T n ( X , L , n ) → Z ~ T ′ n ( X , L , n ) tilde(Z)_(T)^(n)(X,L,n)rarr tilde(Z)_(T^('))^(n)(X,L,n)\tilde{Z}_{T}^{n}(X, \mathscr{L}, n) \rightarrow \tilde{Z}_{T^{\prime}}^{n}(X, \mathscr{L}, n)Z~Tn(X,L,n)→Z~T′n(X,L,n). The rank map G W ( ) Z G W ( − ) → Z GW(-)rarrZ\mathrm{GW}(-) \rightarrow \mathbb{Z}GW(−)→Z gives the homomorphism Z ~ T n ( X , L , n ) Z T n ( X ) Z ~ T n ( X , L , n ) → Z T n ( X ) tilde(Z)_(T)^(n)(X,L,n)rarrZ_(T)^(n)(X)\tilde{Z}_{T}^{n}(X, \mathscr{L}, n) \rightarrow Z_{T}^{n}(X)Z~Tn(X,L,n)→ZTn(X).
Definition 4.12. For X , Y X , Y X,YX, YX,Y in Sm k Sm k Sm_(k)\operatorname{Sm}_{k}Smk, let A ( X , Y ) A ( X , Y ) A(X,Y)\mathcal{A}(X, Y)A(X,Y) be the set of closed subsets T X × Y T ⊂ X × Y T sub X xx YT \subset X \times YT⊂X×Y such that each component of T T TTT is finite over X X XXX and maps surjectively onto an irreducible component of X X XXX. We make A ( X , Y ) A ( X , Y ) A(X,Y)\mathcal{A}(X, Y)A(X,Y) a poset by the inclusion of closed subsets.
Note that if Y Y YYY is irreducible of dimension n n nnn, then a closed subset T X × Y T ⊂ X × Y T sub X xx YT \subset X \times YT⊂X×Y is in A ( X , Y ) A ( X , Y ) A(X,Y)\mathcal{A}(X, Y)A(X,Y) if and only if T T TTT is finite over X X XXX and has pure codimension n n nnn on X × Y X × Y X xx YX \times YX×Y.
Definition 4.13 (Calmès-Fasel [ 31 , § 4.1 ] [ 31 , § 4.1 ] [31,§4.1][31, \S 4.1][31,§4.1] ). Let X , Y X , Y X,YX, YX,Y be in Sm k Sm k Sm_(k)\operatorname{Sm}_{k}Smk and suppose Y Y YYY is irreducible of dimension n n nnn. Define
Corr ~ k ( X , Y ) = colim T A ( X , Y ) Z ~ T n ( X × Y , p 2 ω Y / k ) Corr ~ k ( X , Y ) = colim T ∈ A ( X , Y ) ⁡ Z ~ T n X × Y , p 2 ∗ ω Y / k widetilde(Corr)_(k)(X,Y)=colim_(T inA(X,Y)) tilde(Z)_(T)^(n)(X xx Y,p_(2)^(**)omega_(Y//k))\widetilde{\operatorname{Corr}}_{k}(X, Y)=\operatorname{colim}_{T \in \mathcal{A}(X, Y)} \tilde{Z}_{T}^{n}\left(X \times Y, p_{2}^{*} \omega_{Y / k}\right)Corr~k(X,Y)=colimT∈A(X,Y)⁡Z~Tn(X×Y,p2∗ωY/k)
Extend the definition to general Y Y YYY by additivity.
Using the functorial properties of pullback, intersection product and proper pushforward for the Chow-Witt groups with support, we have a well-defined composition law
Corr ~ k ( Y , Z ) × Corr ~ k ( X , Y ) Corr ~ k ( X , Z ) Corr ~ k ( Y , Z ) × Corr ~ k ( X , Y ) → Corr ~ k ( X , Z ) widetilde(Corr)_(k)(Y,Z)xx widetilde(Corr)_(k)(X,Y)rarr widetilde(Corr)_(k)(X,Z)\widetilde{\operatorname{Corr}}_{k}(Y, Z) \times \widetilde{\operatorname{Corr}}_{k}(X, Y) \rightarrow \widetilde{\operatorname{Corr}}_{k}(X, Z)Corr~k(Y,Z)×Corr~k(X,Y)→Corr~k(X,Z)
via the same formula used to define the composition in C o r k C o r k Cor_(k)\mathrm{Cor}_{k}Cork,
Z 2 Z 1 := p X Z ( p Y Z ( Z 2 ) p X Y ( Z 1 ) ) Z 2 ∘ Z 1 := p X Z ∗ p Y Z ∗ Z 2 ∩ p X Y ∗ Z 1 Z_(2)@Z_(1):=p_(XZ**)(p_(YZ)^(**)(Z_(2))nnp_(XY)^(**)(Z_(1)))Z_{2} \circ Z_{1}:=p_{X Z *}\left(p_{Y Z}^{*}\left(Z_{2}\right) \cap p_{X Y}^{*}\left(Z_{1}\right)\right)Z2∘Z1:=pXZ∗(pYZ∗(Z2)∩pXY∗(Z1))
The twisting by the relative dualizing sheaf in the definition of Corr ~ k ( , ) Corr ~ k ( − , − ) widetilde(Corr)_(k)(-,-)\widetilde{\operatorname{Corr}}_{k}(-,-)Corr~k(−,−) is exactly what is needed for the push-forward map p X Z p X Z ∗ p_(XZ)**p_{X Z} *pXZ∗ to be defined.
This defines the additive category Corr ~ k  Corr  ~ k widetilde(" Corr ")_(k)\widetilde{\text { Corr }}_{k} Corr ~k with objects S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk and morphisms Corr ~ k ( X , Y ) Corr ~ k ( X , Y ) widetilde(Corr)_(k)(X,Y)\widetilde{\operatorname{Corr}}_{k}(X, Y)Corr~k(X,Y). The rank map gives an additive functor
rnk : Corr ~ k Cor k  rnk :  Corr ~ k →  Cor  k " rnk : " widetilde(Corr)_(k)rarr" Cor "_(k)\text { rnk : } \widetilde{\operatorname{Corr}}_{k} \rightarrow \text { Cor }_{k} rnk : Corr~k→ Cor k
One then follows the program used by Voevodsky to define the abelian category of Nisnevich sheaves with Milnor-Witt transfers, Sh Nis M W ( k ) Sh Nis  M W ⁡ ( k ) Sh_("Nis ")^(MW)(k)\operatorname{Sh}_{\text {Nis }}^{\mathrm{MW}}(k)ShNis MW⁡(k), and then D M ~ eff ( k ) D M ~ eff  ( k ) ⊂ widetilde(DM)^("eff ")(k)sub\widetilde{\mathrm{DM}}^{\text {eff }}(k) \subsetDM~eff (k)⊂ D ( Sh N i s M W t r ( k ) ) D Sh N i s M W t r ⁡ ( k ) D(Sh_(Nis)^(MWtr)(k))D\left(\operatorname{Sh}_{\mathrm{Nis}}^{\mathrm{MWtr}}(k)\right)D(ShNisMWtr⁡(k)) as the full subcategory of complexes with strictly A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy invariant cohomology sheaves. One has the localization functor
L ~ A 1 : D ( S h N i s M W t r ( k ) ) D M ~ e f f ( k ) L ~ A 1 : D S h N i s M W t r ( k ) → D M ~ e f f ( k ) tilde(L)_(A^(1)):D(Sh_(Nis)^(MWtr)(k))rarr widetilde(DM)^(eff)(k)\tilde{L}_{\mathbb{A}^{1}}: D\left(\mathrm{Sh}_{\mathrm{Nis}}^{\mathrm{MWtr}}(k)\right) \rightarrow \widetilde{\mathrm{DM}}^{\mathrm{eff}}(k)L~A1:D(ShNisMWtr(k))→DM~eff(k)
constructed using the Suslin complex, the representable sheaves Z ~ t r ( X ) Z ~ t r ( X ) tilde(Z)^(tr)(X)\tilde{\mathbb{Z}}^{\mathrm{tr}}(X)Z~tr(X) for X Sm k X ∈ Sm k X inSm_(k)X \in \operatorname{Sm}_{k}X∈Smk, their corresponding motives M ~ eff ( X ) := L ~ A 1 ( Z ~ tr ( X ) ) D M ~ eff ( k ) M ~ eff  ( X ) := L ~ A 1 Z ~ tr  ( X ) ∈ D M ~ eff  ( k ) tilde(M)^("eff ")(X):= tilde(L)_(A^(1))( tilde(Z)^("tr ")(X))in widetilde(DM)^("eff ")(k)\tilde{M}^{\text {eff }}(X):=\tilde{L}_{\mathbb{A}^{1}}\left(\tilde{Z}^{\text {tr }}(X)\right) \in \widetilde{\mathrm{DM}}^{\text {eff }}(k)M~eff (X):=L~A1(Z~tr (X))∈DM~eff (k) and the Tate motives Z ~ ( n ) Z ~ ( n ) tilde(Z)(n)\tilde{\mathbb{Z}}(n)Z~(n) arising from the reduced motive of P 1 P 1 P^(1)\mathbb{P}^{1}P1. Finally, one constructs D M ~ ( k ) D M ~ ( k ) widetilde(DM)(k)\widetilde{\mathrm{DM}}(k)DM~(k) as a category of Z ~ ( 1 ) Z ~ ( 1 ) tilde(Z)(1)\tilde{\mathbb{Z}}(1)Z~(1) -

M ~ eff ( X ) M ~ eff  ( X ) tilde(M)^("eff ")(X)\tilde{M}^{\text {eff }}(X)M~eff (X).
Definition 4.14. For X S m k X ∈ S m k X inSm_(k)X \in \mathrm{Sm}_{k}X∈Smk, categorical Milnor-Witt cohomology is
H p ( X , Z ~ ( q ) ) := Hom D M ~ ( k ) ( M ~ ( X ) , Z ~ ( q ) [ p ] ) H p ( X , Z ~ ( q ) ) := Hom D M ~ ( k ) ⁡ ( M ~ ( X ) , Z ~ ( q ) [ p ] ) H^(p)(X, tilde(Z)(q)):=Hom_( widetilde(DM)(k))( tilde(M)(X), tilde(Z)(q)[p])H^{p}(X, \tilde{\mathbb{Z}}(q)):=\operatorname{Hom}_{\widetilde{\mathrm{DM}}(k)}(\tilde{M}(X), \tilde{\mathbb{Z}}(q)[p])Hp(X,Z~(q)):=HomDM~(k)⁡(M~(X),Z~(q)[p])
Theorem 4.15. There is a natural isomorphism
H p ( X , Z ~ ( q ) ) H Z ~ p , q ( X ) H p ( X , Z ~ ( q ) ) ≅ H Z ~ p , q ( X ) H^(p)(X, tilde(Z)(q))~=H tilde(Z)^(p,q)(X)H^{p}(X, \tilde{\mathbb{Z}}(q)) \cong H \tilde{\mathbb{Z}}^{p, q}(X)Hp(X,Z~(q))≅HZ~p,q(X)
The proof is very much the same as for motivic cohomology. One shows there is an equivalence of D M ~ ( k ) D M ~ ( k ) widetilde(DM)(k)\widetilde{\mathrm{DM}}(k)DM~(k) with the homotopy category of H Z ~ H Z ~ H tilde(Z)H \tilde{\mathbb{Z}}HZ~-modules (this is [8, THEOREM 5.2]). This gives an adjunction
H Z ~ : S H ( k ) D M ~ ( k ) : E M ~ H Z ~ ∧ − : S H ( k ) ⇄ D M ~ ( k ) : E M ~ H tilde(Z)^^-:SH(k)⇄ widetilde(DM)(k): widetilde(EM)H \tilde{\mathbb{Z}} \wedge-: \mathrm{SH}(k) \rightleftarrows \widetilde{\mathrm{DM}}(k): \widetilde{\mathrm{EM}}HZ~∧−:SH(k)⇄DM~(k):EM~
with H Z ~ H Z ~ ∧ − H tilde(Z)^^-H \tilde{\mathbb{Z}} \wedge-HZ~∧− the free H Z ~ H Z ~ H tilde(Z)H \tilde{\mathbb{Z}}HZ~ module functor and the Eilenberg-MacLane functor E M ~ E M ~ widetilde(EM)\widetilde{\mathrm{EM}}EM~ the forgetful functor. This gives EM ~ ( Z ~ ( 0 ) ) = H Z ~ , M ~ ( X ) = H Z ~ Σ P 1 X + EM ~ ( Z ~ ( 0 ) ) = H Z ~ , M ~ ( X ) = H Z ~ ∧ Σ P 1 ∞ X + widetilde(EM)( tilde(Z)(0))=H tilde(Z), tilde(M)(X)=H tilde(Z)^^Sigma_(P^(1))^(oo)X_(+)\widetilde{\operatorname{EM}}(\tilde{\mathbb{Z}}(0))=H \tilde{\mathbb{Z}}, \tilde{M}(X)=H \tilde{\mathbb{Z}} \wedge \Sigma_{\mathbb{P}^{1}}^{\infty} X_{+}EM~(Z~(0))=HZ~,M~(X)=HZ~∧ΣP1∞X+, and induces the isomorphism
H p ( X , Z ~ ( q ) ) = Hom D M ~ ( k ) ( M ~ ( X ) , Z ~ ( q ) [ p ] ) Hom S H ( k ) ( Σ P 1 X + , Σ p , q H Z ~ ) = H Z ~ p , q ( X ) H p ( X , Z ~ ( q ) ) = Hom D M ~ ( k ) ⁡ ( M ~ ( X ) , Z ~ ( q ) [ p ] ) ≅ Hom S H ( k ) ⁡ Σ P 1 ∞ X + , Σ p , q H Z ~ = H Z ~ p , q ( X ) {:[H^(p)(X"," tilde(Z)(q))=Hom_( widetilde(DM)(k))( tilde(M)(X)"," tilde(Z)(q)[p])],[~=Hom_(SH(k))(Sigma_(P^(1))^(oo)X_(+),Sigma^(p,q)H( tilde(Z)))=H tilde(Z)^(p,q)(X)]:}\begin{aligned} H^{p}(X, \tilde{\mathbb{Z}}(q)) & =\operatorname{Hom}_{\widetilde{\mathrm{DM}}(k)}(\tilde{M}(X), \tilde{\mathbb{Z}}(q)[p]) \\ & \cong \operatorname{Hom}_{\mathrm{SH}(k)}\left(\Sigma_{\mathbb{P}^{1}}^{\infty} X_{+}, \Sigma^{p, q} H \tilde{\mathbb{Z}}\right)=H \tilde{Z}^{p, q}(X) \end{aligned}Hp(X,Z~(q))=HomDM~(k)⁡(M~(X),Z~(q)[p])≅HomSH(k)⁡(ΣP1∞X+,Σp,qHZ~)=HZ~p,q(X)

5. CHOW GROUPS AND MOTIVIC COHOMOLOGY WITH MODULUS

Up to now, all the version of motivic cohomology we have considered share the A 1 A 1 A^(1)\mathbb{A}^{1}A1 homotopy invariance property, namely, that H ( X , Z ( ) ) H ( X × A 1 , Z ( ) ) H ∗ ( X , Z ( ∗ ) ) ≅ H ∗ X × A 1 , Z ( ∗ ) H^(**)(X,Z(**))~=H^(**)(X xxA^(1),Z(**))H^{*}(X, \mathbb{Z}(*)) \cong H^{*}\left(X \times \mathbb{A}^{1}, \mathbb{Z}(*)\right)H∗(X,Z(∗))≅H∗(X×A1,Z(∗)); essentially by construction, this property is enjoyed by all theories that are represented in the motivic stable homotopy category. Although this is a fundamental property controlling a large collection of cohomology theories, this places a serious restriction in at least two naturally occurring areas.
One is the use of deformation theory. This relies on having useful invariants defined on non-reduced schemes, but a cohomology theory that satisfies A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariance will not distinguish between a scheme and its reduced closed subscheme. The second occurs in ramification theory. An A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy invariant theory will not detect Artin-Schreyer covers, and would not give invariants that detect wild ramification.
Fortunately, we have an interesting cohomology theory that is not A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy invariant, namely, algebraic K K KKK-theory, that we can use as a model for a general theory. Algebraic K K KKK-theory does satisfy the A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariance property when restricted to regular schemes, but in general this fails. Besides allowing K K KKK-theory to have a role in deformation theory and ramification theory, this lack of A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariance gives rise to interesting invariants of singularities.

5.1. Higher Chow groups with modulus

The theory of Chow groups with modulus attempts to refine the classical theory of the Chow groups to be useful in both of these areas. This is still a theory in the process of development; just as in the early days of motivic cohomology, many approaches are inspired by properties of algebraic K K KKK-theory.
The tangent space to the functor X O X × X ↦ O X × X|->O_(X)^(xx)X \mapsto \mathcal{O}_{X}^{\times}X↦OX×is given by the structure sheaf, X O X X ↦ O X X|->O_(X)X \mapsto \mathcal{O}_{X}X↦OX, via the isomorphism
O X [ ε ] / ( ε 2 ) × O X × ε O X O X [ ε ] / ε 2 × ≅ O X × ⊕ ε â‹… O X O_(X[epsi]//(epsi^(2)))^(xx)~=O_(X)^(xx)o+epsi*O_(X)\mathcal{O}_{X[\varepsilon] /\left(\varepsilon^{2}\right)}^{\times} \cong \mathcal{O}_{X}^{\times} \oplus \varepsilon \cdot \mathcal{O}_{X}OX[ε]/(ε2)×≅OX×⊕ε⋅OX
Via the isomorphism Pic ( X ) H 1 ( X , O X × ) Pic ⁡ ( X ) ≅ H 1 X , O X × Pic(X)~=H^(1)(X,O_(X)^(xx))\operatorname{Pic}(X) \cong H^{1}\left(X, \mathcal{O}_{X}^{\times}\right)Pic⁡(X)≅H1(X,OX×), this shows that the tangent space at X X XXX to the functor Pic ( ) Pic ⁡ ( − ) Pic(-)\operatorname{Pic}(-)Pic⁡(−) is H 1 ( X , O X ) H 1 X , O X H^(1)(X,O_(X))H^{1}\left(X, \mathcal{O}_{X}\right)H1(X,OX).
In [23], Bloch computes the tangent space to K 2 K 2 K_(2)K_{2}K2 (on local Q Q Q\mathbb{Q}Q-algebras), giving the isomorphism of sheaves on X Z a r X Z a r X_(Zar)X_{\mathrm{Zar}}XZar (for X X XXX a Q Q Q\mathbb{Q}Q-scheme)
K 2 , X [ ε ] / ( ε 2 ) K 2 , X Ω X K 2 , X [ ε ] / ε 2 ≅ K 2 , X ⊕ Ω X K_(2,X[epsi]//(epsi^(2)))~=K_(2,X)o+Omega_(X)\mathcal{K}_{2, X[\varepsilon] /\left(\varepsilon^{2}\right)} \cong \mathcal{K}_{2, X} \oplus \Omega_{X}K2,X[ε]/(ε2)≅K2,X⊕ΩX
where Ω X Ω X Omega_(X)\Omega_{X}ΩX is the sheaf of absolute Kähler differentials. Bloch then uses his formula from [22],
H 2 ( X Z a r , K 2 ) C H 2 ( X ) H 2 X Z a r , K 2 ≅ C H 2 ( X ) H^(2)(X_(Zar),K_(2))~=CH^(2)(X)H^{2}\left(X_{\mathrm{Zar}}, \mathcal{K}_{2}\right) \cong \mathrm{CH}^{2}(X)H2(XZar,K2)≅CH2(X)
to justify defining C H 2 ( X [ ε ] / ( ε 2 ) ) C H 2 X [ ε ] / ε 2 CH^(2)(X[epsi]//(epsi^(2)))\mathrm{CH}^{2}\left(X[\varepsilon] /\left(\varepsilon^{2}\right)\right)CH2(X[ε]/(ε2)) as H 2 ( X [ ε ] / ( ε 2 ) Z a r , K 2 ) H 2 X [ ε ] / ε 2 Z a r , K 2 H^(2)(X[epsi]//(epsi^(2))_(Zar),K_(2))H^{2}\left(X[\varepsilon] /\left(\varepsilon^{2}\right)_{\mathrm{Zar}}, \mathcal{K}_{2}\right)H2(X[ε]/(ε2)Zar,K2), giving
C H 2 ( X [ ε ] / ( ε 2 ) ) = C H 2 ( X ) H 2 ( X , Ω X ) C H 2 X [ ε ] / ε 2 = C H 2 ( X ) ⊕ H 2 X , Ω X CH^(2)(X[epsi]//(epsi^(2)))=CH^(2)(X)o+H^(2)(X,Omega_(X))\mathrm{CH}^{2}\left(X[\varepsilon] /\left(\varepsilon^{2}\right)\right)=\mathrm{CH}^{2}(X) \oplus H^{2}\left(X, \Omega_{X}\right)CH2(X[ε]/(ε2))=CH2(X)⊕H2(X,ΩX)
For X X XXX a smooth projective surface over C C C\mathbb{C}C with H 2 ( X , O X ) 0 H 2 X , O X ≠ 0 H^(2)(X,O_(X))!=0H^{2}\left(X, \mathcal{O}_{X}\right) \neq 0H2(X,OX)≠0, the exact sheaf sequence
0 Ω C / Q C O X Ω X Ω X / C 0 0 → Ω C / Q ⊗ C O X → Ω X → Ω X / C → 0 0rarrOmega_(C//Q)ox_(C)O_(X)rarrOmega_(X)rarrOmega_(X//C)rarr00 \rightarrow \Omega_{\mathbb{C} / \mathbb{Q}} \otimes_{\mathbb{C}} \mathcal{O}_{X} \rightarrow \Omega_{X} \rightarrow \Omega_{X / \mathbb{C}} \rightarrow 00→ΩC/Q⊗COX→ΩX→ΩX/C→0
along the fact that Ω C / Q Ω C / Q Omega_(C//Q)\Omega_{\mathbb{C} / \mathbb{Q}}ΩC/Q is a C C C\mathbb{C}C-vector space of uncountable dimension show that Ω C / Q C H 2 ( X , O X ) Ω C / Q ⊗ C H 2 X , O X Omega_(C//Q)ox_(C)H^(2)(X,O_(X))\Omega_{\mathbb{C} / \mathbb{Q}} \otimes_{\mathbb{C}} H^{2}\left(X, \mathcal{O}_{X}\right)ΩC/Q⊗CH2(X,OX) makes a huge contribution to the tangent space H 2 ( X , Ω X ) H 2 X , Ω X H^(2)(X,Omega_(X))H^{2}\left(X, \Omega_{X}\right)H2(X,ΩX) of C H 2 ( ) C H 2 ( − ) CH^(2)(-)\mathrm{CH}^{2}(-)CH2(−) on X X XXX. This is reflected in Mumford's result [95], that if H 2 ( X , O X ) H 0 ( X , Ω X / C 2 ) H 2 X , O X ≅ H 0 X , Ω X / C 2 H^(2)(X,O_(X))~=H^(0)(X,Omega_(X//C)^(2))H^{2}\left(X, \mathcal{O}_{X}\right) \cong H^{0}\left(X, \Omega_{X / \mathbb{C}}^{2}\right)H2(X,OX)≅H0(X,ΩX/C2) is nonzero, then C H 2 ( X ) C H 2 ( X ) CH^(2)(X)\mathrm{CH}^{2}(X)CH2(X) is "infinite-dimensional," and gives some evidence for Bloch's conjecture [23, CONJECTURE (0.4)] on 0 -cycles on surfaces X X XXX with H 0 ( X , Ω X / C 2 ) = 0 H 0 X , Ω X / C 2 = 0 H^(0)(X,Omega_(X//C)^(2))=0H^{0}\left(X, \Omega_{X / \mathbb{C}}^{2}\right)=0H0(X,ΩX/C2)=0.
The algebraic cycles have disappeared in this approach to Chow groups of nonreduced schemes. Bloch and Esnault [26] gave the first construction of a cycle-theoretic theory that could say something interesting about higher cycles on the non-reduced scheme Spec k [ ε ] / ( ε 2 ) k [ ε ] / ε 2 k[epsi]//(epsi^(2))k[\varepsilon] /\left(\varepsilon^{2}\right)k[ε]/(ε2). In a second paper [27], they modified and extended this construction to give a theory of additive higher Chow groups with modulus m m mmm, for the field k k kkk. This was motivated by Bloch's earlier use of K K KKK-theory on the affine line, relative to { 0 , 1 } { 0 , 1 } {0,1}\{0,1\}{0,1}, to study K 3 K 3 K_(3)K_{3}K3. Letting 1 tend to 0 , they were led to consider the relative K K KKK-theory space K ( k [ ε ] , ( ε 2 ) ) K k [ ε ] , ε 2 K(k[epsi],(epsi^(2)))K\left(k[\varepsilon],\left(\varepsilon^{2}\right)\right)K(k[ε],(ε2)), this being the homotopy fiber of the restriction map K ( k [ ε ] ) K ( k [ ε ] / ε 2 ) K ( k [ ε ] ) → K k [ ε ] / ε 2 K(k[epsi])rarr K(k[epsi]//epsi^(2))K(k[\varepsilon]) \rightarrow K\left(k[\varepsilon] / \varepsilon^{2}\right)K(k[ε])→K(k[ε]/ε2), whose homotopy groups are the relative K K KKK-theory groups K n ( k [ ε ] , ( ε 2 ) ) K n k [ ε ] , ε 2 K_(n)(k[epsi],(epsi^(2)))K_{n}\left(k[\varepsilon],\left(\varepsilon^{2}\right)\right)Kn(k[ε],(ε2)). Replacing 2 with m 2 m ≥ 2 m >= 2m \geq 2m≥2 gives the relative K K KKK-theory groups K n ( k [ ε ] , ( ε m ) ) K n k [ ε ] , ε m K_(n)(k[epsi],(epsi^(m)))K_{n}\left(k[\varepsilon],\left(\varepsilon^{m}\right)\right)Kn(k[ε],(εm)). This led to the consideration of a complex of cycles on Spec k [ ε ] k [ ε ] k[epsi]k[\varepsilon]k[ε], with an additional condition imposed on the m m mmm th order limiting behavior of the cycles; an explicit construction of such a cycle complex with modulus, z q ( k , , m ) z q ( k , ∗ , m ) z^(q)(k,**,m)z^{q}(k, *, m)zq(k,∗,m) was given in [27]. The homology A C H q ( k , p , m ) := H p ( z q ( k , , m ) ) A C H q ( k , p , m ) := H p z q ( k , ∗ , m ) ACH^(q)(k,p,m):=H_(p)(z^(q)(k,**,m))A \mathrm{CH}^{q}(k, p, m):=H_{p}\left(z^{q}(k, *, m)\right)ACHq(k,p,m):=Hp(zq(k,∗,m)) defines the additive codimension
q q qqq higher Chow groups with modulus m m mmm for Spec k k kkk. Bloch-Esnault recover the computation A C H n ( k , n 1 , 2 ) Ω k n 1 A C H n ( k , n − 1 , 2 ) ≅ Ω k n − 1 ACH^(n)(k,n-1,2)~=Omega_(k)^(n-1)A \mathrm{CH}^{n}(k, n-1,2) \cong \Omega_{k}^{n-1}ACHn(k,n−1,2)≅Ωkn−1 from [26], and relate the additive analogue of weight two K 3 K 3 K_(3)K_{3}K3, A C H 2 ( k , 2 , 2 ) A C H 2 ( k , 2 , 2 ) ACH^(2)(k,2,2)A \mathrm{CH}^{2}(k, 2,2)ACH2(k,2,2), with the additive dilogarithm of Cathelineau [32].
Rülling [104] studied the projective system
A C H n ( k , n 1 , m + 1 ) A C H n ( k , n 1 , m ) ⋯ → A C H n ( k , n − 1 , m + 1 ) → A C H n ( k , n − 1 , m ) → ⋯ cdots rarr ACH^(n)(k,n-1,m+1)rarr ACH^(n)(k,n-1,m)rarr cdots\cdots \rightarrow A \mathrm{CH}^{n}(k, n-1, m+1) \rightarrow A \mathrm{CH}^{n}(k, n-1, m) \rightarrow \cdots⋯→ACHn(k,n−1,m+1)→ACHn(k,n−1,m)→⋯
He showed this is endowed with additional endomorphisms F n F n F_(n)F_{n}Fn and V n V n V_(n)V_{n}Vn, and the graded group n A C H n ( k , n 1 , + 1 ) 2 ⨁ n   A C H n ( k , n − 1 , ∗ + 1 ) ∗ ≥ 2 bigoplus_(n)ACH^(n)(k,n-1,**+1)_(** >= 2)\bigoplus_{n} A \mathrm{CH}^{n}(k, n-1, *+1)_{* \geq 2}⨁nACHn(k,n−1,∗+1)∗≥2 has the structure of a pro-differential graded algebra. In fact, we have
Theorem 5.1 (Rülling). Let k k kkk be a field of characteristic 2 ≠ 2 !=2\neq 2≠2. The pro-dga n A C H n ( k ⨁ n   A C H n ( k bigoplus_(n)ACH^(n)(k\bigoplus_{n} A \mathrm{CH}^{n}(k⨁nACHn(k, n 1 , + 1 ) n − 1 , ∗ + 1 ) n-1,**+1)n-1, *+1)n−1,∗+1), with F n F n F_(n)F_{n}Fn as Frobenius and V n V n V_(n)V_{n}Vn as Verschiebung, is isomorphic to the de RhamWitt complex of Madsen-Hesselholt,
n A C H n ( k , n 1 , + 1 ) n W Ω k n 1 ⨁ n   A C H n ( k , n − 1 , ∗ + 1 ) ≅ ⨁ n   W ∗ Ω k n − 1 bigoplus_(n)ACH^(n)(k,n-1,**+1)~=bigoplus_(n)W_(**)Omega_(k)^(n-1)\bigoplus_{n} A \mathrm{CH}^{n}(k, n-1, *+1) \cong \bigoplus_{n} W_{*} \Omega_{k}^{n-1}⨁nACHn(k,n−1,∗+1)≅⨁nW∗Ωkn−1
With essentially the same definition as given by Bloch-Esnault, the additive cycle complex and additive Chow groups were extended to arbitrary k k kkk-schemes Y Y YYY by Park [98], replacing A 1 A 1 A^(1)\mathbb{A}^{1}A1 and divisor m 0 m â‹… 0 m*0m \cdot 0mâ‹…0 with the scheme Y × A 1 Y × A 1 Y xxA^(1)Y \times \mathbb{A}^{1}Y×A1 and divisor m Y × 0 m â‹… Y × 0 m*Y xx0m \cdot Y \times 0mâ‹…Y×0. Binda and Saito [20] went one step further, defining complexes z q ( X , D , ) z q ( X , D , ∗ ) z^(q)(X,D,**)z^{q}(X, D, *)zq(X,D,∗) for a pair ( X , D ) ( X , D ) (X,D)(X, D)(X,D) of a finite type separated k k kkk-scheme X X XXX and a Cartier divisor D D DDD, using essentially the same definition as before. The homology is the higher Chow group with modulus
C H q ( X , D , p ) := H p ( z q ( X , D , ) ) C H q ( X , D , p ) := H p z q ( X , D , ∗ ) CH^(q)(X,D,p):=H_(p)(z^(q)(X,D,**))\mathrm{CH}^{q}(X, D, p):=H_{p}\left(z^{q}(X, D, *)\right)CHq(X,D,p):=Hp(zq(X,D,∗))
The constructions of Bloch-Esnault, Park, and Binda-Saito all use a cubical model of Bloch's cycle complex. Here one replaces the algebraic n n nnn-simplex, Δ k n = Spec k [ t 0 , Δ k n = Spec ⁡ k t 0 , … Delta_(k)^(n)=Spec k[t_(0),dots:}\Delta_{k}^{n}=\operatorname{Spec} k\left[t_{0}, \ldots\right.Δkn=Spec⁡k[t0,…, t n ] / i t i 1 t n / ∑ i   t i − 1 {:t_(n)]//sum_(i)t_(i)-1\left.t_{n}\right] / \sum_{i} t_{i}-1tn]/∑iti−1, with the algebraic n n nnn-cube
n := ( P 1 { 1 } , 0 , ) n ◻ n := P 1 ∖ { 1 } , 0 , ∞ n ◻^(n):=(P^(1)\\{1},0,oo)^(n)\square^{n}:=\left(\mathbb{P}^{1} \backslash\{1\}, 0, \infty\right)^{n}◻n:=(P1∖{1},0,∞)n
The notation means that one considers ( P 1 { 1 } ) n A n P 1 ∖ { 1 } n ≅ A n (P^(1)\\{1})^(n)~=A^(n)\left(\mathbb{P}^{1} \backslash\{1\}\right)^{n} \cong \mathbb{A}^{n}(P1∖{1})n≅An with its "faces" defined by setting some of the coordinates equal to 0 or ∞ oo\infty∞. The corresponding cycle complex z q ( X , ) c z q ( X , ∗ ) c z^(q)(X,**)_(c)z^{q}(X, *)_{c}zq(X,∗)c has degree n n nnn component z q ( X , n ) c z q ( X , n ) c z^(q)(X,n)_(c)z^{q}(X, n)_{c}zq(X,n)c the codimension q q qqq cycles on X × n X × â—» n X xxâ—»^(n)X \times \square^{n}X×◻n that intersect X × F X × F X xx FX \times FX×F properly for all faces F F FFF of n â—» n â—»^(n)\square^{n}â—»n; one also needs to quotient out by the degenerate cycles, these being the ones that come by pullback via projection to a m â—» m â—»^(m)\square^{m}â—»m with m < n m < n m < nm<nm<n. The differential is again an alternating sum of restrictions to the maximal faces t i = 0 t i = 0 t_(i)=0t_{i}=0ti=0 and t i = t i = ∞ t_(i)=oot_{i}=\inftyti=∞.
This complex also computes the motivic cohomology of X X XXX, just as Bloch's simplicial cycle complex does. In the Binda-Saito construction, the modulus condition arises by considering the closed box ¯ n := ( P 1 ) n â—» ¯ n := P 1 n bar(â—»)^(n):=(P^(1))^(n)\bar{\square}^{n}:=\left(\mathbb{P}^{1}\right)^{n}◻¯n:=(P1)n. Let F n i ( P 1 ) n F n i ⊂ P 1 n F_(n)^(i)sub(P^(1))^(n)F_{n}^{i} \subset\left(\mathbb{P}^{1}\right)^{n}Fni⊂(P1)n be the divisor defined by t i = 1 t i = 1 t_(i)=1t_{i}=1ti=1 and let F n = i = 1 n F n i F n = ∑ i = 1 n   F n i F_(n)=sum_(i=1)^(n)F_(n)^(i)F_{n}=\sum_{i=1}^{n} F_{n}^{i}Fn=∑i=1nFni. In ( P 1 ) n × X P 1 n × X (P^(1))^(n)xx X\left(\mathbb{P}^{1}\right)^{n} \times X(P1)n×X we have two distinguished Cartier divisors, ( P 1 ) n × D P 1 n × D (P^(1))^(n)xx D\left(\mathbb{P}^{1}\right)^{n} \times D(P1)n×D and F n × X F n × X F_(n)xx XF_{n} \times XFn×X. A subvariety Z ( P 1 { 1 } ) n × X Z ⊂ P 1 ∖ { 1 } n × X Z sub(P^(1)\\{1})^(n)xx XZ \subset\left(\mathbb{P}^{1} \backslash\{1\}\right)^{n} \times XZ⊂(P1∖{1})n×X that is in z q ( X , n ) c z q ( X , n ) c z^(q)(X,n)_(c)z^{q}(X, n)_{c}zq(X,n)c satisfies the modulus condition if
p ( F n × X ) p ( ( P 1 ) n × D ) p ∗ F n × X ≥ p ∗ P 1 n × D p^(**)(F_(n)xx X) >= p^(**)((P^(1))^(n)xx D)p^{*}\left(F_{n} \times X\right) \geq p^{*}\left(\left(\mathbb{P}^{1}\right)^{n} \times D\right)p∗(Fn×X)≥p∗((P1)n×D)
where p : Z ¯ N ( P 1 ) n × X p : Z ¯ N → P 1 n × X p: bar(Z)^(N)rarr(P^(1))^(n)xx Xp: \bar{Z}^{N} \rightarrow\left(\mathbb{P}^{1}\right)^{n} \times Xp:Z¯N→(P1)n×X is the normalization of the closure of Z Z ZZZ in ¯ n × X â—» ¯ n × X bar(â—»)^(n)xx X\bar{\square}^{n} \times X◻¯n×X. Restricting to the subgroup of Z q ( n × X ) Z q â—» n × X Z^(q)(â—»^(n)xx X)Z^{q}\left(\square^{n} \times X\right)Zq(â—»n×X) generated by codimension q q qqq subvarieties Z n × X Z ⊂ â—» n × X Z subâ—»^(n)xx XZ \subset \square^{n} \times XZ⊂◻n×X that intersect faces properly and satisfy the modulus condition yields the cycle complex with modulus z q ( X ; D , ) z q ( X , ) c z q ( X ; D , ∗ ) ⊂ z q ( X , ∗ ) c z^(q)(X;D,**)subz^(q)(X,**)_(c)z^{q}(X ; D, *) \subset z^{q}(X, *)_{c}zq(X;D,∗)⊂zq(X,∗)c; the higher Chow groups with modulus is then defined as
C H q ( X ; D , p ) := H p ( z q ( X ; D , ) ) C H q ( X ; D , p ) := H p z q ( X ; D , ∗ ) CH^(q)(X;D,p):=H_(p)(z^(q)(X;D,**))\mathrm{CH}^{q}(X ; D, p):=H_{p}\left(z^{q}(X ; D, *)\right)CHq(X;D,p):=Hp(zq(X;D,∗))
The second construction of Bloch-Esnault, and Park's generalization, are recovered as the special cases X = A k 1 X = A k 1 X=A_(k)^(1)X=\mathbb{A}_{k}^{1}X=Ak1 and D = m 0 D = m â‹… 0 D=m*0D=m \cdot 0D=mâ‹…0 in the Bloch-Esnault version and X = Y × A 1 X = Y × A 1 X=Y xxA^(1)X=Y \times \mathbb{A}^{1}X=Y×A1, D = m Y × 0 D = m â‹… Y × 0 D=m*Y xx0D=m \cdot Y \times 0D=mâ‹…Y×0 in Park's version.
For X X XXX a finite type k k kkk-scheme, recall the Bloch motivic complex Z B I ( q ) X Z B I ( q ) X ∗ Z_(BI)(q)_(X)^(**)\mathbb{Z}_{\mathrm{BI}}(q)_{X}^{*}ZBI(q)X∗ defined as the Zariski sheafification of the presheaf U z q ( X , 2 q ) U ↦ z q ( X , 2 q − ∗ ) U|->z^(q)(X,2q-**)U \mapsto z^{q}(X, 2 q-*)U↦zq(X,2q−∗) (this is already a Nisnevich sheaf). Bloch's cycle complexes satisfy an important localization property: the natural maps to Zariski and Nisnevich hypercohomology
H p ( z q ( X , 2 q ) ) H p ( X Z a r , Z B l ( q ) X ) H p ( X N i s , Z B l ( q ) X ) H p z q ( X , 2 q − ∗ ) → H p X Z a r , Z B l ( q ) X ∗ → H p X N i s , Z B l ( q ) X ∗ H^(p)(z^(q)(X,2q-**))rarrH^(p)(X_(Zar),Z_(Bl)(q)_(X)^(**))rarrH^(p)(X_(Nis),Z_(Bl)(q)_(X)^(**))H^{p}\left(z^{q}(X, 2 q-*)\right) \rightarrow \mathbb{H}^{p}\left(X_{\mathrm{Zar}}, \mathbb{Z}_{\mathrm{Bl}}(q)_{X}^{*}\right) \rightarrow \mathbb{H}^{p}\left(X_{\mathrm{Nis}}, \mathbb{Z}_{\mathrm{Bl}}(q)_{X}^{*}\right)Hp(zq(X,2q−∗))→Hp(XZar,ZBl(q)X∗)→Hp(XNis,ZBl(q)X∗)
are isomorphisms. This fails for the cycle complex with modulus, although the comparison between the Zariski and Nisnevich hypercohomology seems to be still an open question.
Iwasa and Kai consider the Nisnevich sheafification Z ( q ) ( X ; D ) Z ( q ) ( X ; D ) ∗ Z(q)_((X;D))^(**)\mathcal{Z}(q)_{(X ; D)}^{*}Z(q)(X;D)∗ of the presheaf
U z q ( U ; D × X U , 2 q ) U ↦ z q U ; D × X U , 2 q − ∗ U|->z^(q)(U;Dxx_(X)U,2q-**)U \mapsto z^{q}\left(U ; D \times_{X} U, 2 q-*\right)U↦zq(U;D×XU,2q−∗)
We call H p ( X N i s , Z ( q ) ( X ; D ) ) H p X N i s , Z ( q ) ( X ; D ) ∗ H^(p)(X_(Nis),Z(q)_((X;D))^(**))\mathbb{H}^{p}\left(X_{\mathrm{Nis}}, \mathcal{Z}(q)_{(X ; D)}^{*}\right)Hp(XNis,Z(q)(X;D)∗) the motivic cohomology with modulus for ( X , D ) ( X , D ) (X,D)(X, D)(X,D). Kai [74] shows that this sheafified version has contravariant functoriality. Iwasa and Kai [67] construct Chern class maps from relative K K KKK-theory
c p , q : K 2 q p ( X ; D ) H p ( X N i s , Z ( q ) X , N i s ) c p , q : K 2 q − p ( X ; D ) → H p X N i s , Z ( q ) X , N i s ∗ c_(p,q):K_(2q-p)(X;D)rarrH^(p)(X_(Nis),Z(q)_(X,Nis)^(**))c_{p, q}: K_{2 q-p}(X ; D) \rightarrow \mathbb{H}^{p}\left(X_{\mathrm{Nis}}, \mathcal{Z}(q)_{X, \mathrm{Nis}}^{*}\right)cp,q:K2q−p(X;D)→Hp(XNis,Z(q)X,Nis∗)

5.2. 0 -cycles with modulus and class field theory

There is a classical theory of 0 -cycles on a smooth complete curve C C CCC with a modulus condition at a finite set of points S S SSS, due to Rosenlicht and Serre [109, III]. The idea is quite simple, instead of relations coming from divisors (zeros minus poles) of an arbitrary rational function f , f f , f f,ff, ff,f is required to have a power series expansion at each point p S p ∈ S p in Sp \in Sp∈S, with leading term 1 and the next nonzero term of the form u t p n p u t p n p ut_(p)^(n_(p))u t_{p}^{n_{p}}utpnp, with u ( p ) 0 , t p u ( p ) ≠ 0 , t p u(p)!=0,t_(p)u(p) \neq 0, t_{p}u(p)≠0,tp a local coordinate at p p ppp and the integer n p > 0 n p > 0 n_(p) > 0n_{p}>0np>0 being the "modulus." This is applied to the class field theory of a smooth open curve U C U ⊂ C U sub CU \subset CU⊂C over a finite field [109, THEOREM 4], that identifies the inverse limit of the groups of degree 0 cycle classes on U U UUU, with modulus supported in C U C ∖ U C\\UC \backslash UC∖U, with the kernel of the map π 1 et ( U ) a b Gal ( k ¯ / k ) Ï€ 1 et  ( U ) a b → Gal ⁡ ( k ¯ / k ) pi_(1)^("et ")(U)^(ab)rarr Gal( bar(k)//k)\pi_{1}^{\text {et }}(U)^{a b} \rightarrow \operatorname{Gal}(\bar{k} / k)Ï€1et (U)ab→Gal⁡(k¯/k).
In their class field theory for higher-dimensional varieties, Kato and Saito [80] introduce a group of 0 -cycles on a k k kkk-scheme X X XXX with modulus D D DDD, defined by
C H 0 ( X , D ) := H n ( X , K n , ( X , D ) M ) C H 0 ( X , D ) := H n X , K n , ( X , D ) M CH_(0)(X,D):=H^(n)(X,K_(n,(X,D))^(M))\mathrm{CH}_{0}(X, D):=H^{n}\left(X, \mathcal{K}_{n,(X, D)}^{M}\right)CH0(X,D):=Hn(X,Kn,(X,D)M)
with K n , ( X , D ) M K n , ( X , D ) M K_(n,(X,D))^(M)\mathcal{K}_{n,(X, D)}^{M}Kn,(X,D)M a relative version of the Milnor K K KKK-theory sheaf, recalling Kato's isomorphism H n ( X , K n M ) C H n ( X ) H n X , K n M ≅ C H n ( X ) H^(n)(X,K_(n)^(M))~=CH^(n)(X)H^{n}\left(X, \mathcal{K}_{n}^{M}\right) \cong \mathrm{CH}^{n}(X)Hn(X,KnM)≅CHn(X) for X X XXX a smooth k k kkk-scheme [78]. Kerz and Saito give a different definition of a group of relative 0 -cycles C ( X , D ) C ( X , D ) C(X,D)C(X, D)C(X,D) on a normal k k kkk-scheme X X XXX with effective Cartier
divisor D D DDD such that X D X ∖ D X\\DX \backslash DX∖D is smooth. It follows from their comments in [81, DEFINITION 1.6] that C ( X , D ) = C H n ( X ; D , 0 ) C ( X , D ) = C H n ( X ; D , 0 ) C(X,D)=CH^(n)(X;D,0)C(X, D)=\mathrm{CH}^{n}(X ; D, 0)C(X,D)=CHn(X;D,0) for X X XXX of dimension n n nnn, and it is easy to see that the KatoSaito and Kerz-Saito relative 0 -cycles agree with the Rosenlicht-Serre groups in the case of curves.
Kerz and Saito consider a smooth finite-type k k kkk-scheme U U UUU, choose a normal compactification X X XXX and define the topological group C ( U ) := lim D C ( X , D ) C ( U ) := lim D   C ( X , D ) C(U):=lim_(D)C(X,D)C(U):=\lim _{D} C(X, D)C(U):=limDC(X,D), as D D DDD runs over effective Cartier divisors on X X XXX, supported in X U X ∖ U X\\UX \backslash UX∖U, and with each C ( X , D ) C ( X , D ) C(X,D)C(X, D)C(X,D) given the discrete topology. They show that C ( U ) C ( U ) C(U)C(U)C(U) is independent of the choice of X X XXX, and their main result generalizes class field theory for smooth curves over a finite field as described above.
Theorem 5.2 ([81, Ñ‚HEORem 3.3]). Let k k kkk be a finite field of characteristic 2 ≠ 2 !=2\neq 2≠2 and let U U UUU be a smooth variety over k k kkk, Then C ( U ) C ( U ) C(U)C(U)C(U) is isomorphic as topological group to a dense subgroup of the abelianized étale fundamental group π 1 e t ( U ) a b Ï€ 1 e t ( U ) a b pi_(1)^(et)(U)^(ab)\pi_{1}^{e t}(U)^{a b}Ï€1et(U)ab and this isomorphism induces an isomorphism of the degree 0 part C ( U ) 0 C ( U ) 0 C(U)^(0)C(U)^{0}C(U)0 of C ( U ) C ( U ) C(U)C(U)C(U) with the kernel π 1 e t ( U ) 0 a b Ï€ 1 e t ( U ) 0 a b pi_(1)^(et)(U)_(0)^(ab)\pi_{1}^{e t}(U)_{0}^{a b}Ï€1et(U)0ab of π 1 e t ( U ) a b π 1 e t ( k ) Ï€ 1 e t ( U ) a b → Ï€ 1 e t ( k ) pi_(1)^(et)(U)^(ab)rarrpi_(1)^(et)(k)\pi_{1}^{e t}(U)^{a b} \rightarrow \pi_{1}^{e t}(k)Ï€1et(U)ab→π1et(k).

5.3. Categories of motives with modulus

There has been a great deal of interest in constructing a categorical framework for motivic cohomology with modulus. A central issue is the lack of A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy invariance for this theory, which raises the question of what type of homotopy invariance should replace this.
One direction has been the construction of a reasonable replacement for the category of homotopy invariant Nisnevich sheaves with transfers. A non-homotopy invariant version has been developed via the theory of reciprocity sheaves, the name coming from the reciprocity laws in class field theory of curves and its relation to the group of 0 -cycles with modulus of Rosenlicht-Serre. We will say a bit about reciprocity sheaves later on, in the context of motives for log schemes Section 5.4.
For now, we will look at categories of motives with modulus constructed on the Voevodsky model by introducing a new notion of correspondence and a suitable replacement for A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy invariance.
Looking at algebraic K K KKK-theory, the closest replacement for A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy invariance seems to be the P 1 P 1 P^(1)\mathbb{P}^{1}P1-bundle formula
K n ( X × P 1 ) K ( X ) [ O X × P 1 ] K ( X ) [ O X × P 1 ( 1 ) ] K n X × P 1 ≅ K ( X ) â‹… O X × P 1 ⊕ K ( X ) â‹… O X × P 1 ( − 1 ) K_(n)(X xxP^(1))~=K(X)*[O_(X xxP^(1))]o+K(X)*[O_(X xxP^(1))(-1)]K_{n}\left(X \times \mathbb{P}^{1}\right) \cong K(X) \cdot\left[\mathcal{O}_{X \times \mathbb{P}^{1}}\right] \oplus K(X) \cdot\left[\mathcal{O}_{X \times \mathbb{P}^{1}}(-1)\right]Kn(X×P1)≅K(X)â‹…[OX×P1]⊕K(X)â‹…[OX×P1(−1)]
valid for a general scheme X X XXX. This has led to attempts to create a category of motives with
Here one has the problem that P 1 P 1 P^(1)\mathbb{P}^{1}P1 does not have the structure of an interval, a structure enjoyed by A 1 A 1 A^(1)\mathbb{A}^{1}A1. One considers A 1 A 1 A^(1)\mathbb{A}^{1}A1 together with "endpoints" 0,1 . Following the general theory of a site with interval, as developed by Morel-Voevodsky [94, chap. 2], one needs the multiplication map m : A 1 × A 1 A 1 m : A 1 × A 1 → A 1 m:A^(1)xxA^(1)rarrA^(1)m: \mathbb{A}^{1} \times \mathbb{A}^{1} \rightarrow \mathbb{A}^{1}m:A1×A1→A1 to allow one to consider ( A 1 , 0 , 1 ) A 1 , 0 , 1 (A^(1),0,1)\left(\mathbb{A}^{1}, 0,1\right)(A1,0,1) as an abstract interval. In the construction of the cycle complex with modulus, one identifies ( A 1 , 0 , 1 ) A 1 , 0 , 1 (A^(1),0,1)\left(\mathbb{A}^{1}, 0,1\right)(A1,0,1) with ( P 1 { 1 } , 0 , ) P 1 ∖ { 1 } , 0 , ∞ (P^(1)\\{1},0,oo)\left(\mathbb{P}^{1} \backslash\{1\}, 0, \infty\right)(P1∖{1},0,∞), and the corresponding multiplication map m : ( P 1 { 1 } ) × ( P 1 { 1 } ) m ′ : P 1 ∖ { 1 } × P 1 ∖ { 1 } → m^('):(P^(1)\\{1})xx(P^(1)\\{1})rarrm^{\prime}:\left(\mathbb{P}^{1} \backslash\{1\}\right) \times\left(\mathbb{P}^{1} \backslash\{1\}\right) \rightarrowm′:(P1∖{1})×(P1∖{1})→ P 1 { 1 } P 1 ∖ { 1 } P^(1)\\{1}\mathbb{P}^{1} \backslash\{1\}P1∖{1} only extends as a rational map P 1 × P 1 P 1 P 1 × P 1 → P 1 P^(1)xxP^(1)rarrP^(1)\mathbb{P}^{1} \times \mathbb{P}^{1} \rightarrow \mathbb{P}^{1}P1×P1→P1. However, m m ′ m^(')m^{\prime}m′ becomes a morphism
after blowing up the point ( 1 , 1 ) ( 1 , 1 ) (1,1)(1,1)(1,1), which suggests that one should consider the closure of the graph of m m ′ m^(')m^{\prime}m′ in P 1 × P 1 × P 1 P 1 × P 1 × P 1 P^(1)xxP^(1)xxP^(1)\mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{1}P1×P1×P1 as an allowable correspondence from P 1 × P 1 P 1 × P 1 P^(1)xxP^(1)\mathbb{P}^{1} \times \mathbb{P}^{1}P1×P1 to P 1 P 1 P^(1)\mathbb{P}^{1}P1.
With this as starting point, Kahn, Miyazaki, Saito, and Yamazaki [69-71] follow Voevodsky's program, defining a category of modulus correspondences MCor k k _(k)_{k}k. Objects are pairs ( M ¯ , M M ¯ , M ∞ bar(M),M^(oo)\bar{M}, M^{\infty}M¯,M∞ ) with M ¯ M ¯ bar(M)\bar{M}M¯ a separated finite-type k k kkk-scheme and M M ∞ M^(oo)M^{\infty}M∞ an effective Cartier divisor on M ¯ M ¯ bar(M)\bar{M}M¯ such that the open complement M := M ¯ M M ∘ := M ¯ ∖ M ∞ M^(@):= bar(M)\\M^(oo)M^{\circ}:=\bar{M} \backslash M^{\infty}M∘:=M¯∖M∞ is smooth. The morphism group

Z Z ZZZ (finite and surjective over a component of M M ∘ M^(@)M^{\circ}M∘ ) such that
(i) The closure Z ¯ Z ¯ bar(Z)\bar{Z}Z¯ of Z Z ZZZ in M ¯ × N ¯ M ¯ × N ¯ bar(M)xx bar(N)\bar{M} \times \bar{N}M¯×N¯ is proper over M ¯ M ¯ bar(M)\bar{M}M¯ (not necessarily finite).
(ii) Let f : Z ¯ N M ¯ × N ¯ f : Z ¯ N → M ¯ × N ¯ f: bar(Z)^(N)rarr bar(M)xx bar(N)f: \bar{Z}^{N} \rightarrow \bar{M} \times \bar{N}f:Z¯N→M¯×N¯ be the normalization of Z ¯ Z ¯ bar(Z)\bar{Z}Z¯. Then
f p 1 M f p 2 N f ∗ p 1 ∗ M ∞ ≥ f ∗ p 2 ∗ N ∞ f^(**)p_(1)^(**)M^(oo) >= f^(**)p_(2)^(**)N^(oo)f^{*} p_{1}^{*} M^{\infty} \geq f^{*} p_{2}^{*} N^{\infty}f∗p1∗M∞≥f∗p2∗N∞
The composition law in C o r k C o r k Cor_(k)\mathrm{Cor}_{k}Cork preserves conditions (i) and (ii), giving the category M C o r _ k M C o r _ k MCor__(k)\underline{M C o r}_{k}MCor_k with functor M _ Cor k Cor k M _ Cor k → Cor k M_Cor_(k)rarrCor_(k)\underline{\mathrm{M}} \operatorname{Cor}_{k} \rightarrow \operatorname{Cor}_{k}M_Cork→Cork sending ( M ¯ , M ) M ¯ , M ∞ (( bar(M)),M^(oo))\left(\bar{M}, M^{\infty}\right)(M¯,M∞) to M M ∘ M^(@)M^{\circ}M∘ and with M _ Cor k ( ( M ¯ , M ) M _ Cor k ⁡ M ¯ , M ∞ M_Cor_(k)((( bar(M)),M^(oo)):}\underline{\operatorname{M}} \operatorname{Cor}_{k}\left(\left(\bar{M}, M^{\infty}\right)\right.M_Cork⁡((M¯,M∞), ( N ¯ , N ) ) Cor k ( M , N ) N ¯ , N ∞ → Cor k ⁡ M ∘ , N ∘ {:(( bar(N)),N^(oo)))rarrCor_(k)(M^(@),N^(@))\left.\left(\bar{N}, N^{\infty}\right)\right) \rightarrow \operatorname{Cor}_{k}\left(M^{\circ}, N^{\circ}\right)(N¯,N∞))→Cork⁡(M∘,N∘) the inclusion. The product of pairs makes M C o r k _ M C o r k _ MCor_(k)_\underline{\mathrm{MCor}_{k}}MCork_ a symmetric monoidal category and the functor to C o r k C o r k Cor_(k)\mathrm{Cor}_{k}Cork is symmetric monoidal.
Let ¯ â—» ¯ bar(â—»)\bar{\square}◻¯ be the object ( P 1 , { 1 } ) P 1 , { 1 } (P^(1),{1})\left(\mathbb{P}^{1},\{1\}\right)(P1,{1}). As hinted above, the closure of the graph of m : ( P 1 { 1 } ) × ( P 1 { 1 } ) P 1 { 1 } m ′ : P 1 ∖ { 1 } × P 1 ∖ { 1 } → P 1 ∖ { 1 } m^('):(P^(1)\\{1})xx(P^(1)\\{1})rarrP^(1)\\{1}m^{\prime}:\left(\mathbb{P}^{1} \backslash\{1\}\right) \times\left(\mathbb{P}^{1} \backslash\{1\}\right) \rightarrow \mathbb{P}^{1} \backslash\{1\}m′:(P1∖{1})×(P1∖{1})→P1∖{1} defines a morphism m : ¯ × ¯ ¯ m : â—» ¯ × â—» ¯ → â—» ¯ m: bar(â—»)xx bar(â—»)rarr bar(â—»)m: \bar{\square} \times \bar{\square} \rightarrow \bar{\square}m:◻¯×◻¯→◻¯ in MCor k k _(k)_{k}k.
They then consider the abelian category of additive presheaves of abelian groups

pairs ( X , D ) ( X , D ) (X,D)(X, D)(X,D), with X X XXX a proper k k kkk-scheme, as a full subcategory of MCor _ k  MCor  _ k " MCor "__(k)\underline{\text { MCor }}{ }_{k} MCor _k, with its presheaf category MPST k MPST k MPST_(k)\operatorname{MPST}_{k}MPSTk.
They define a category of effective proper motives with modulus, MDM e f f ( k ) MDM e f f ⁡ ( k ) MDM^(eff)(k)\operatorname{MDM}^{\mathrm{eff}}(k)MDMeff⁡(k), by localizing the derived category D ( MPST k ) D MPST k D(MPST_(k))D\left(\operatorname{MPST}_{k}\right)D(MPSTk). Roughly speaking, they follow the Voevodsky program, replacing the A 1 A 1 A^(1)\mathbb{A}^{1}A1-localization with ¯ â—» ¯ bar(â—»)\bar{\square}◻¯ localization. To get the proper Nisnevich localization is a bit technical; we refer the reader to [71, DEFINITION 1.3.9] for details.
There is still quite a bit that is not known. One central problem is how to realize the various constructions of the higher Chow groups with modulus as morphisms in a suitable triangulated category. There is a connection, at least for the modulus version of Suslin homology and the Suslin complex, which we now describe.
One can show that the cubical version of the Suslin complex
C Sus ( X ) c ( Y ) := Hom Cor k ( Y × , X ) / degn C ∗ Sus  ( X ) c ( Y ) := Hom Cor  k ⁡ Y × â—» ∗ , X /  degn  C_(**)^("Sus ")(X)_(c)(Y):=Hom_("Cor "_(k))(Y xxâ—»^(**),X)//" degn "C_{*}^{\text {Sus }}(X)_{c}(Y):=\operatorname{Hom}_{\text {Cor }_{k}}\left(Y \times \square^{*}, X\right) / \text { degn }C∗Sus (X)c(Y):=HomCor k⁡(Y×◻∗,X)/ degn 
is naturally quasi-isomorphic to the simplicial version C Sus ( X ) ( Y ) C ∗ Sus  ( X ) ( Y ) C_(**)^("Sus ")(X)(Y)C_{*}^{\text {Sus }}(X)(Y)C∗Sus (X)(Y), where / degn means taking the quotient by the image of the pullback maps via the projections Y × n Y × Y × â—» n → Y × Y xxâ—»^(n)rarr Y xxY \times \square^{n} \rightarrow Y \timesY×◻n→Y× n 1 â—» n − 1 â—»^(n-1)\square^{n-1}â—»n−1. For a modulus pair ( X , D ) ( X , D ) (X,D)(X, D)(X,D), one can similarly form the naive Suslin complex
C Sus ( X , D ) ( Y , E ) := Hom M C o r _ k ( ( Y , E ) ¯ , X ) / degn C ∗ Sus  ( X , D ) ( Y , E ) := Hom M C o r _ k ⁡ ( Y , E ) ⊗ â—» ¯ ∗ , X /  degn  C_(**)^("Sus ")(X,D)(Y,E):=Hom_(MCor__(k))((Y,E)ox bar(â—»)^(**),X)//" degn "C_{*}^{\text {Sus }}(X, D)(Y, E):=\operatorname{Hom}_{\underline{M C o r}_{k}}\left((Y, E) \otimes \bar{\square}^{*}, X\right) / \text { degn }C∗Sus (X,D)(Y,E):=HomMCor_k⁡((Y,E)⊗◻¯∗,X)/ degn 
Taking ( Y , E ) = ( Spec k , ) ( Y , E ) = ( Spec ⁡ k , ∅ ) (Y,E)=(Spec k,O/)(Y, E)=(\operatorname{Spec} k, \emptyset)(Y,E)=(Spec⁡k,∅), we have the complex
C S u s ( X , D ) := C S u s ( X , D ) ( Spec k , ) C ∗ S u s ( X , D ) := C ∗ S u s ( X , D ) ( Spec ⁡ k , ∅ ) C_(**)^(Sus)(X,D):=C_(**)^(Sus)(X,D)(Spec k,O/)C_{*}^{\mathrm{Sus}}(X, D):=C_{*}^{\mathrm{Sus}}(X, D)(\operatorname{Spec} k, \emptyset)C∗Sus(X,D):=C∗Sus(X,D)(Spec⁡k,∅)
Next, there is a derived Suslin complex R C Sus ( X , D ) c ( ) R C ∗ Sus  ( X , D ) c ( − ) RC_(**)^("Sus ")(X,D)_(c)(-)R C_{*}^{\text {Sus }}(X, D)_{c}(-)RC∗Sus (X,D)c(−) with a natural map of presheaves
C S u s ( X , D ) c ( ) R C S u s ( X , D ) c ( ) C ∗ S u s ( X , D ) c ( − ) → R C ∗ S u s ( X , D ) c ( − ) C_(**)^(Sus)(X,D)_(c)(-)rarr RC_(**)^(Sus)(X,D)_(c)(-)C_{*}^{\mathrm{Sus}}(X, D)_{c}(-) \rightarrow R C_{*}^{\mathrm{Sus}}(X, D)_{c}(-)C∗Sus(X,D)c(−)→RC∗Sus(X,D)c(−)
By [71, THEOREM 2], for ( X , D ) ( X , D ) (X,D)(X, D)(X,D) a proper modulus pair, R C Sus ( X , D ) c ( ) R C ∗ Sus  ( X , D ) c ( − ) RC_(**)^("Sus ")(X,D)_(c)(-)R C_{*}^{\text {Sus }}(X, D)_{c}(-)RC∗Sus (X,D)c(−) computes the maps in MDM e f f ( k ) MDM e f f ⁡ ( k ) MDM^(eff)(k)\operatorname{MDM}^{\mathrm{eff}}(k)MDMeff⁡(k) as
H n ( R C Sus ( X , D ) c ( Spec k , ) ) = Hom M D M eff ( k ) ( ( Spec k , ) , M eff ( X , D ) ) H n R C ∗ Sus  ( X , D ) c ( Spec ⁡ k , ∅ ) = Hom M D M eff  ( k ) ⁡ ( Spec ⁡ k , ∅ ) , M eff  ( X , D ) H_(n)(RC_(**)^("Sus ")(X,D)_(c)(Spec k,O/))=Hom_(MDM^("eff ")(k))((Spec k,O/),M^("eff ")(X,D))H_{n}\left(R C_{*}^{\text {Sus }}(X, D)_{c}(\operatorname{Spec} k, \emptyset)\right)=\operatorname{Hom}_{\mathrm{MDM}^{\text {eff }}(k)}\left((\operatorname{Spec} k, \emptyset), M^{\text {eff }}(X, D)\right)Hn(RC∗Sus (X,D)c(Spec⁡k,∅))=HomMDMeff (k)⁡((Spec⁡k,∅),Meff (X,D))
However, one should not expect that the Suslin complex or its derived version should yield a version of the higher Chow groups. If one looks back at the setting of D M ( k ) D M ( k ) DM(k)\mathrm{DM}(k)DM(k), the object that most naturally yields the higher Chow groups for an arbitrary finite type k k kkk-scheme X X XXX is the motive with compact supports M c ( X ) M c ( X ) M^(c)(X)M^{c}(X)Mc(X). This is defined as C Sus ( Z t r c ( X ) ) C ∗ Sus  Z t r c ( X ) C_(**)^("Sus ")(Z_(tr)^(c)(X))C_{*}^{\text {Sus }}\left(\mathbb{Z}_{\mathrm{tr}}^{c}(X)\right)C∗Sus (Ztrc(X)), where Z t r c ( X ) Z t r c ( X ) Z_(tr)^(c)(X)\mathbb{Z}_{\mathrm{tr}}^{c}(X)Ztrc(X) is the presheaf with transfers with Z t r c ( X ) ( Y ) Z t r c ( X ) ( Y ) Z_(tr)^(c)(X)(Y)\mathbb{Z}_{\mathrm{tr}}^{c}(X)(Y)Ztrc(X)(Y) the free abelian group on integral W Y × X W ⊂ Y × X W sub Y xx XW \subset Y \times XW⊂Y×X, with W Y W → Y W rarr YW \rightarrow YW→Y quasi-finite and dominant over a component of Y Sm k Y ∈ Sm k Y inSm_(k)Y \in \operatorname{Sm}_{k}Y∈Smk. See [127, cHAP. 5, PROPOSITION 4.2.9] for the relation of M c ( X ) M c ( X ) M^(c)(X)M^{c}(X)Mc(X) with Bloch's higher Chow groups.
One can define a similar version with modulus as the object M c ( X , D ) M c ( X , D ) M^(c)(X,D)M^{c}(X, D)Mc(X,D) associated to the presheaf Z t r c ( X , D ) Z t r c ( X , D ) Z_(tr)^(c)(X,D)\mathbb{Z}_{\mathrm{tr}}^{c}(X, D)Ztrc(X,D), with Z t r c ( X , D ) ( Y , E ) Z dim Y ( Y × X ) Z t r c ( X , D ) ( Y , E ) ⊂ Z dim ⁡ Y ( Y × X ) Z_(tr)^(c)(X,D)(Y,E)subZ_(dim Y)(Y xx X)\mathbb{Z}_{\mathrm{tr}}^{c}(X, D)(Y, E) \subset Z_{\operatorname{dim} Y}(Y \times X)Ztrc(X,D)(Y,E)⊂Zdim⁡Y(Y×X) the subgroup generated by closed subvarieties W ( Y E ) × ( X D ) W ⊂ ( Y ∖ E ) × ( X ∖ D ) W sub(Y\\E)xx(X\\D)W \subset(Y \backslash E) \times(X \backslash D)W⊂(Y∖E)×(X∖D) that are quasi-finite and dominant over Y Y YYY, and with the usual modulus condition, that the normalization ν : W ¯ N ν : W ¯ N → nu: bar(W)^(N)rarr\nu: \bar{W}^{N} \rightarrowν:W¯N→ Y × X Y × X Y xx XY \times XY×X of the closure of W W WWW in Y × X Y × X Y xx XY \times XY×X satisfies
v ( E × X ) v ( Y × D ) v ∗ ( E × X ) ≥ v ∗ ( Y × D ) v^(**)(E xx X) >= v^(**)(Y xx D)v^{*}(E \times X) \geq v^{*}(Y \times D)v∗(E×X)≥v∗(Y×D)
There is an analog of Suslin's comparison theorem in the affine case, due to KaiMiyazaki [75]: They define an equi-dimensional cycle complex with modulus
z d equi ( X , D , ) z d ( X , D , ) z d equi  ( X , D , ∗ ) ⊂ z d ( X , D , ∗ ) z_(d)^("equi ")(X,D,**)subz_(d)(X,D,**)z_{d}^{\text {equi }}(X, D, *) \subset z_{d}(X, D, *)zdequi (X,D,∗)⊂zd(X,D,∗)
which for d = 0 d = 0 d=0d=0d=0 is the Suslin complex with modulus C Sus ( Z t r c ( X , D ) ) ( Spec k , ) C ∗ Sus  Z t r c ( X , D ) ( Spec ⁡ k , ∅ ) C_(**)^("Sus ")(Z_(tr)^(c)(X,D))(Spec k,O/)C_{*}^{\text {Sus }}\left(\mathbb{Z}_{\mathrm{tr}}^{c}(X, D)\right)(\operatorname{Spec} k, \emptyset)C∗Sus (Ztrc(X,D))(Spec⁡k,∅)
Theorem 5.3 (Kai-Miyazaki). Let ( X , D ) ( X , D ) (X,D)(X, D)(X,D) be a modulus pair, with X X XXX affine. Then there is a pro-isomorphism
{ H ( z d equi ( X , m D , ) ) } m { C H d ( X , m D , ) } m H ∗ z d equi  ( X , m D , ∗ ) m ≅ C H d ( X , m D , ∗ ) m {H_(**)(z_(d)^("equi ")(X,mD,**))}_(m)~={CH_(d)(X,mD,**)}_(m)\left\{H_{*}\left(z_{d}^{\text {equi }}(X, m D, *)\right)\right\}_{m} \cong\left\{\mathrm{CH}_{d}(X, m D, *)\right\}_{m}{H∗(zdequi (X,mD,∗))}m≅{CHd(X,mD,∗)}m
Miyazaki [91] has defined objects z equi ( X , D , d ) MNST k z equi  ( X , D , d ) ∈ MNST k z^("equi ")(X,D,d)inMNST_(k)z^{\text {equi }}(X, D, d) \in \operatorname{MNST}_{k}zequi (X,D,d)∈MNSTk, with Z tr c ( X , D ) = Z tr  c ( X , D ) = Z_("tr ")^(c)(X,D)=\mathbb{Z}_{\text {tr }}^{c}(X, D)=Ztr c(X,D)= z equi ( X , D , 0 ) z equi  ( X , D , 0 ) z^("equi ")(X,D,0)z^{\text {equi }}(X, D, 0)zequi (X,D,0). The sheaf z equi ( X , D , r ) z equi  ( X , D , r ) z^("equi ")(X,D,r)z^{\text {equi }}(X, D, r)zequi (X,D,r) is defined similarly to Z t r c ( X , D ) Z t r c ( X , D ) Z_(tr)^(c)(X,D)\mathbb{Z}_{\mathrm{tr}}^{c}(X, D)Ztrc(X,D), with z equi ( X , D z equi  ( X , D z^("equi ")(X,Dz^{\text {equi }}(X, Dzequi (X,D, d ) ( Y , E ) d ) ( Y , E ) d)(Y,E)d)(Y, E)d)(Y,E) the group of cycles on ( Y E ) × ( X D ) ( Y ∖ E ) × ( X ∖ D ) (Y\\E)xx(X\\D)(Y \backslash E) \times(X \backslash D)(Y∖E)×(X∖D) generated by closed, integral W ( Y E ) × ( X D ) W ⊂ ( Y ∖ E ) × ( X ∖ D ) W sub(Y\\E)xx(X\\D)W \subset(Y \backslash E) \times(X \backslash D)W⊂(Y∖E)×(X∖D) that are equi-dimensional of dimension d d ddd over Y E Y ∖ E Y\\EY \backslash EY∖E, dominate a component of Y E Y ∖ E Y\\EY \backslash EY∖E, and with v : W ¯ N Y × X v : W ¯ N → Y × X v: bar(W)^(N)rarr Y xx Xv: \bar{W}^{N} \rightarrow Y \times Xv:W¯N→Y×X satisfying the modulus condition
v ( E × X ) v ( Y × D ) v ∗ ( E × X ) ≥ v ∗ ( Y × D ) v^(**)(E xx X) >= v^(**)(Y xx D)v^{*}(E \times X) \geq v^{*}(Y \times D)v∗(E×X)≥v∗(Y×D)
Moreover, for an arbitrary modulus pair ( X , D ) ( X , D ) (X,D)(X, D)(X,D), one has
z d equi ( X , D , ) = C Sus ( z equi ( X , D , d ) ) ( Spec k , ) z d equi  ( X , D , ∗ ) = C ∗ Sus  z equi  ( X , D , d ) ( Spec ⁡ k , ∅ ) z_(d)^("equi ")(X,D,**)=C_(**)^("Sus ")(z^("equi ")(X,D,d))(Spec k,O/)z_{d}^{\text {equi }}(X, D, *)=C_{*}^{\text {Sus }}\left(z^{\text {equi }}(X, D, d)\right)(\operatorname{Spec} k, \emptyset)zdequi (X,D,∗)=C∗Sus (zequi (X,D,d))(Spec⁡k,∅)
and there is the canonical map
C Sus ( z e q u i ( X , D , d ) ) R C Sus ( z equi ( X , D , d ) ) C ∗ Sus  z e q u i ( X , D , d ) → R C ∗ Sus  z equi  ( X , D , d ) C_(**)^("Sus ")(z^(equi)(X,D,d))rarr RC_(**)^("Sus ")(z^("equi ")(X,D,d))C_{*}^{\text {Sus }}\left(z^{\mathrm{equi}}(X, D, d)\right) \rightarrow R C_{*}^{\text {Sus }}\left(z^{\text {equi }}(X, D, d)\right)C∗Sus (zequi(X,D,d))→RC∗Sus (zequi (X,D,d))
Letting C H q equi ( X , D , p ) = H p ( z q equi ( X , D , ) ) C H q equi  ( X , D , p ) = H p z q equi  ( X , D , ∗ ) CH_(q)^("equi ")(X,D,p)=H_(p)(z_(q)^("equi ")(X,D,**))\mathrm{CH}_{q}^{\text {equi }}(X, D, p)=H_{p}\left(z_{q}^{\text {equi }}(X, D, *)\right)CHqequi (X,D,p)=Hp(zqequi (X,D,∗)), we have the natural map
C H q equi ( X , D , p ) C H q ( X , D , p ) C H q equi  ( X , D , p ) → C H q ( X , D , p ) CH_(q)^("equi ")(X,D,p)rarrCH_(q)(X,D,p)\mathrm{CH}_{q}^{\text {equi }}(X, D, p) \rightarrow \mathrm{CH}_{q}(X, D, p)CHqequi (X,D,p)→CHq(X,D,p)
which is an isomorphism for X X XXX affine, and we have the natural maps for ( X , D ) ( X , D ) (X,D)(X, D)(X,D) a proper modulus pair
C H q equi ( X , D , p ) H p ( R C Sus ( z equi ( X , D , q ) ) ( Spec k , ) ) Hom MDM eff ( k ) ( M ( Spec k , ) [ p ] , R C Sus ( z equi ( X , D , q ) ) ) C H q equi  ( X , D , p ) → H p R C ∗ Sus  z equi  ( X , D , q ) ( Spec ⁡ k , ∅ ) → Hom MDM eff  ⁡ ( k ) ⁡ M ( Spec ⁡ k , ∅ ) [ p ] , R C ∗ Sus  z equi  ( X , D , q ) {:[CH_(q)^("equi ")(X","D","p) rarrH_(p)(RC_(**)^("Sus ")(z^("equi ")(X,D,q))(Spec k,O/))],[ rarrHom_(MDM^("eff ")(k))(M(Spec k,O/)[p],RC_(**)^("Sus ")(z^("equi ")(X,D,q)))]:}\begin{aligned} \mathrm{CH}_{q}^{\text {equi }}(X, D, p) & \rightarrow H_{p}\left(R C_{*}^{\text {Sus }}\left(z^{\text {equi }}(X, D, q)\right)(\operatorname{Spec} k, \emptyset)\right) \\ & \rightarrow \operatorname{Hom}_{\operatorname{MDM}^{\text {eff }}(k)}\left(M(\operatorname{Spec} k, \emptyset)[p], R C_{*}^{\text {Sus }}\left(z^{\text {equi }}(X, D, q)\right)\right) \end{aligned}CHqequi (X,D,p)→Hp(RC∗Sus (zequi (X,D,q))(Spec⁡k,∅))→HomMDMeff ⁡(k)⁡(M(Spec⁡k,∅)[p],RC∗Sus (zequi (X,D,q)))
For a proper modulus pair, let M c ( X , D ) M c ( X , D ) M^(c)(X,D)M^{c}(X, D)Mc(X,D) denote the image of Z tr c ( X , D ) Z tr  c ( X , D ) Z_("tr ")^(c)(X,D)\mathbb{Z}_{\text {tr }}^{c}(X, D)Ztr c(X,D) in MDM e f f ( k ) MDM e f f ⁡ ( k ) MDM^(eff)(k)\operatorname{MDM}^{\mathrm{eff}}(k)MDMeff⁡(k). One can ask if there are analogs of the theorem of Kahn-Miyazaki-Saito-Yamazaki.
Question 5.4. For ( X , D ) ( X , D ) (X,D)(X, D)(X,D) a proper modulus pair, are the maps
H p ( R C S u s ( Z t r c ( X , D ) ) ( Spec k , ) ) Hom M D M e f f ( k ) ( M ( Spec k , ) [ p ] , M c ( X , D ) ) H p R C ∗ S u s Z t r c ( X , D ) ( Spec ⁡ k , ∅ ) → Hom M D M e f f ( k ) ⁡ M ( Spec ⁡ k , ∅ ) [ p ] , M c ( X , D ) H_(p)(RC_(**)^(Sus)(Z_(tr)^(c)(X,D))(Spec k,O/))rarrHom_(MDM^(eff)(k))(M(Spec k,O/)[p],M^(c)(X,D))H_{p}\left(R C_{*}^{\mathrm{Sus}}\left(\mathbb{Z}_{t r}^{c}(X, D)\right)(\operatorname{Spec} k, \emptyset)\right) \rightarrow \operatorname{Hom}_{\mathrm{MDM}^{\mathrm{eff}}(k)}\left(M(\operatorname{Spec} k, \emptyset)[p], M^{c}(X, D)\right)Hp(RC∗Sus(Ztrc(X,D))(Spec⁡k,∅))→HomMDMeff(k)⁡(M(Spec⁡k,∅)[p],Mc(X,D))
isomorphisms? More generally, are the maps
H p ( R C Sus ( z equi ( X , D , q ) ) ( Spec k , ) ) Hom MDM eff ( k ) ( M ( Spec k , ) [ p ] , R C Sus ( z equi ( X , D , q ) ) ) H p R C ∗ Sus  z equi  ( X , D , q ) ( Spec ⁡ k , ∅ ) → Hom MDM eff  ⁡ ( k ) ⁡ M ( Spec ⁡ k , ∅ ) [ p ] , R C ∗ Sus  z equi  ( X , D , q ) {:[H_(p)(RC_(**)^("Sus ")(z^("equi ")(X,D,q))(Spec k,O/))],[quad rarrHom_(MDM^("eff ")(k))(M(Spec k,O/)[p],RC_(**)^("Sus ")(z^("equi ")(X,D,q)))]:}\begin{aligned} & H_{p}\left(R C_{*}^{\text {Sus }}\left(z^{\text {equi }}(X, D, q)\right)(\operatorname{Spec} k, \emptyset)\right) \\ & \quad \rightarrow \operatorname{Hom}_{\operatorname{MDM}^{\text {eff }}(k)}\left(M(\operatorname{Spec} k, \emptyset)[p], R C_{*}^{\text {Sus }}\left(z^{\text {equi }}(X, D, q)\right)\right) \end{aligned}Hp(RC∗Sus (zequi (X,D,q))(Spec⁡k,∅))→HomMDMeff ⁡(k)⁡(M(Spec⁡k,∅)[p],RC∗Sus (zequi (X,D,q)))
isomorphisms?
It is also not clear if the map
C H q equi ( X , D , p ) H p ( R C Sus ( z equi ( X , D , q ) ) ( Spec k , ) ) C H q equi  ( X , D , p ) → H p R C ∗ Sus  z equi  ( X , D , q ) ( Spec ⁡ k , ∅ ) CH_(q)^("equi ")(X,D,p)rarrH_(p)(RC_(**)^("Sus ")(z^("equi ")(X,D,q))(Spec k,O/))\mathrm{CH}_{q}^{\text {equi }}(X, D, p) \rightarrow H_{p}\left(R C_{*}^{\text {Sus }}\left(z^{\text {equi }}(X, D, q)\right)(\operatorname{Spec} k, \emptyset)\right)CHqequi (X,D,p)→Hp(RC∗Sus (zequi (X,D,q))(Spec⁡k,∅))
should be an isomorphism. Possibly one should also consider the Nisnevich hypercohomology H p ( X N i s , Z q equi ( X , D , ) ) H − p X N i s , Z q equi  ( X , D , ∗ ) H^(-p)(X_(Nis),Z_(q)^("equi ")(X,D,**))\mathbb{H}^{-p}\left(X_{\mathrm{Nis}}, \mathcal{Z}_{q}^{\text {equi }}(X, D, *)\right)H−p(XNis,Zqequi (X,D,∗)), with Z q equi ( X , D , ) Z q equi  ( X , D , ∗ ) Z_(q)^("equi ")(X,D,**)\mathcal{Z}_{q}^{\text {equi }}(X, D, *)Zqequi (X,D,∗) defined by sheafifying U U ↦ U|->U \mapstoU↦ z q equi ( U , U D , ) z q equi  ( U , U ∩ D , ∗ ) z_(q)^("equi ")(U,U nn D,**)z_{q}^{\text {equi }}(U, U \cap D, *)zqequi (U,U∩D,∗).
For Voevodsky motives, and for X X XXX a finite type k k kkk-scheme, the motivic Borel-Moore homology is defined by
H p B.M. ( X , Z ( q ) ) := Hom D M eff ( k ) ( Z ( q ) [ p ] , M ( X ) c ) H p 2 q ( z q equi ( X , ) ) H p 2 q ( z q ( X , ) ) = C H q ( X , p 2 q ) H p B.M.  ( X , Z ( q ) ) := Hom D M eff  ( k ) ⁡ Z ( q ) [ p ] , M ( X ) c ≅ H p − 2 q z q equi  ( X , ∗ ) ≅ H p − 2 q z q ( X , ∗ ) = C H q ( X , p − 2 q ) {:[H_(p)^("B.M. ")(X","Z(q)):=Hom_(DM^("eff ")(k))(Z(q)[p],M(X)^(c))],[~=H_(p-2q)(z_(q)^("equi ")(X,**))~=H_(p-2q)(z_(q)(X,**))=CH_(q)(X","p-2q)]:}\begin{aligned} H_{p}^{\text {B.M. }}(X, \mathbb{Z}(q)) & :=\operatorname{Hom}_{\mathrm{DM}^{\text {eff }}(k)}\left(\mathbb{Z}(q)[p], M(X)^{c}\right) \\ & \cong H_{p-2 q}\left(z_{q}^{\text {equi }}(X, *)\right) \cong H_{p-2 q}\left(z_{q}(X, *)\right)=\mathrm{CH}_{q}(X, p-2 q) \end{aligned}HpB.M. (X,Z(q)):=HomDMeff (k)⁡(Z(q)[p],M(X)c)≅Hp−2q(zqequi (X,∗))≅Hp−2q(zq(X,∗))=CHq(X,p−2q)
This uses the duality M ( X ) c M ( X ) ( d ) [ 2 d ] M ( X ) c ≅ M ( X ) ∨ ( d ) [ 2 d ] M(X)^(c)~=M(X)^(vv)(d)[2d]M(X)^{c} \cong M(X)^{\vee}(d)[2 d]M(X)c≅M(X)∨(d)[2d] for X X XXX of dimension d d ddd (valid in characteristic zero, or after inverting p p ppp in characteristic p > 0 p > 0 p > 0p>0p>0 ), and the extension of Suslin's quasi-isomorphism z q equi ( X , ) z q ( X , ) z q equi  ( X , ∗ ) ↪ z q ( X , ∗ ) z_(q)^("equi ")(X,**)↪z_(q)(X,**)z_{q}^{\text {equi }}(X, *) \hookrightarrow z_{q}(X, *)zqequi (X,∗)↪zq(X,∗) to arbitrary X X XXX. Moreover, we have M ( X ) c = M ( X ) c = M(X)^(c)=M(X)^{c}=M(X)c= M ( X ) M ( X ) M(X)M(X)M(X) for X X XXX smooth and proper.
However, a corresponding motivic cohomology of modulus pairs seems to need a larger category. This is hinted at by the use of the duality (in DM ( k ) ) M ( X ) c DM ⁡ ( k ) ) M ( X ) c ≅ DM(k))M(X)^(c)~=\operatorname{DM}(k)) M(X)^{c} \congDM⁡(k))M(X)c≅ M ( X ) ( d ) [ 2 d ] M ( X ) ∨ ( d ) [ 2 d ] M(X)^(vv)(d)[2d]M(X)^{\vee}(d)[2 d]M(X)∨(d)[2d] in the computations described above. This says in particular that each motive M ( X ) M ( X ) M(X)M(X)M(X) admits a "twisted" dual in D M e f f ( k ) D M e f f ( k ) DM^(eff)(k)\mathrm{DM}^{\mathrm{eff}}(k)DMeff(k), more precisely, the usual evaluation
and coevaluation maps associated with a dual exist, but as maps with target or source some Z ( d ) [ 2 d ] Z ( d ) [ 2 d ] Z(d)[2d]\mathbb{Z}(d)[2 d]Z(d)[2d] rather than the unit. For a general proper modulus pair ( X , D ) ( X , D ) (X,D)(X, D)(X,D), this does not seem to be the case; one seems to need modulus pairs with an anti-effective Cartier divisor. Another way to say the same thing, if one looks for a proper modulus pair ( X , D ) X , D ′ (X,D^('))\left(X, D^{\prime}\right)(X,D′) such that Hom MDM eff ( k ) ( M ( X , D ) , Z ( q ) [ p ] ) Hom MDM eff  ⁡ ( k ) ⁡ M X , D ′ , Z ( q ) [ p ] Hom_(MDM^("eff ")(k))(M(X,D^(')),Z(q)[p])\operatorname{Hom}_{\operatorname{MDM}^{\text {eff }}(k)}\left(M\left(X, D^{\prime}\right), \mathbb{Z}(q)[p]\right)HomMDMeff ⁡(k)⁡(M(X,D′),Z(q)[p]) looks at all like C H q ( X , D , 2 q p ) C H q ( X , D , 2 q − p ) CH^(q)(X,D,2q-p)\mathrm{CH}^{q}(X, D, 2 q-p)CHq(X,D,2q−p) for some given proper modulus pair ( X , D ) ( X , D ) (X,D)(X, D)(X,D), the defining inequalities in Corr k k _(k){ }_{k}k suggest that D D ′ D^(')D^{\prime}D′ could be D − D -D-D−D. See the section "Perspectives" in [71, InTRoduction] for further details in this direction.

5.4. Logarithmic motives and reciprocity sheaves

Grothendieck motives for log schemes have been constructed in [66], where a version for mixed motives has also been constructed using systems of realizations. There the emphasis is on versions of motives for homological or numerical equivalence in the setting of log log log\loglog schemes. In this section we discuss a recent construction of a triangulated category of log motives, by Binda-Park-Østvær [19], that follows the Voevodsky program. We refer the reader to the lectures notes of Ogus [97] for the facts about log schemes.
Recall that a log log log\loglog scheme is a pair ( X , α : M ( O X , × ) ) X , α : M → O X , × (X,alpha:Mrarr(O_(X),xx))\left(X, \alpha: \mathcal{M} \rightarrow\left(\mathcal{O}_{X}, \times\right)\right)(X,α:M→(OX,×)) consisting of a scheme X X XXX and a homomorphism of sheaves of commutative monoids α : M ( O X , × ) α : M → O X , × alpha:Mrarr(O_(X),xx)\alpha: \mathcal{M} \rightarrow\left(\mathcal{O}_{X}, \times\right)α:M→(OX,×) such that α 1 ( O X × ) O X × Î± − 1 O X × → O X × alpha^(-1)(O_(X)^(xx))rarrO_(X)^(xx)\alpha^{-1}\left(\mathcal{O}_{X}^{\times}\right) \rightarrow \mathcal{O}_{X}^{\times}α−1(OX×)→OX×is an isomorphism; without this last condition, the pair ( X , α : M ( X , α : M → (X,alpha:Mrarr(X, \alpha: \mathcal{M} \rightarrow(X,α:M→ ( O X , × ) ) O X , × {:(O_(X),xx))\left.\left(\mathcal{O}_{X}, \times\right)\right)(OX,×)) is called a pre-log structure. A pre-log structure α : M ( O X , × ) α : M → O X , × alpha:Mrarr(O_(X),xx)\alpha: \mathcal{M} \rightarrow\left(\mathcal{O}_{X}, \times\right)α:M→(OX,×) induces a log structure α log : M log ( O X , × ) α log : M log → O X , × alpha^(log):M^(log)rarr(O_(X),xx)\alpha^{\log }: \mathcal{M}^{\log } \rightarrow\left(\mathcal{O}_{X}, \times\right)αlog:Mlog→(OX,×) by taking M log M log M^(log)\mathcal{M}^{\log }Mlog to be the push-out (in the category of sheaves of monoids) in
Given a modulus pair ( X , D ) ( X , D ) (X,D)(X, D)(X,D), there are a number of (in general distinct) induced log structures on X X XXX. For example, one can take the compactifying log structure, with M := O X M := O X ∩ M:=O_(X)nn\mathcal{M}:=\mathcal{O}_{X} \capM:=OX∩ j O U × j ∗ O U × j_(**)O_(U)^(xx)j_{*} \mathcal{O}_{U}^{\times}j∗OU×, where U = X D U = X ∖ D U=X\\DU=X \backslash DU=X∖D and j : U X j : U → X j:U rarr Xj: U \rightarrow Xj:U→X is the inclusion. There are other log structures, which in general depend on a choice of decomposition of D D DDD as a sum of effective Cartier divisors (for example, the Deligne-Faltings log structure, discussed in [97, III, DEFINITION 1.7.1]).
Replacing the category of smooth k k kkk-schemes is the category 1 S m k 1 S m k 1Sm_(k)1 \mathrm{Sm}_{k}1Smk of fine, saturated, log smooth and separated log schemes over the log scheme Spec k k kkk endowed with the trivial log log log\loglog structure. We refer the reader to [19] for details; one needs these technical conditions to construct the category of finite log correspondences. We call a separated, fine, saturated log scheme an fs log scheme.
We sketch the construction of the category of finite log correspondences, and describe how Binda-Park- stvær follow Voevodsky's program to define the triangulated category log D M eff ( k ) log ⁡ D M eff  ( k ) log DM^("eff ")(k)\log \mathrm{DM}^{\text {eff }}(k)log⁡DMeff (k) of effective log motives over k k kkk.
For X Smm k X ∈ Smm k X inSmm_(k)X \in \operatorname{Smm}_{k}X∈Smmk, let X _ X _ X_\underline{X}X_ denote the underlying k k kkk-scheme. We let X X _ X ∘ ⊂ X _ X^(@)subX_X^{\circ} \subset \underline{X}X∘⊂X_ denote the maximal open subscheme over which the log structure M X O X M X → O X M_(X)rarrO_(X)\mathcal{M}_{X} \rightarrow \mathcal{O}_{X}MX→OX is trivial, that is, M X U = O U × M X ∣ U = O U × M_(X∣U)=O_(U)^(xx)\mathcal{M}_{X \mid U}=\mathcal{O}_{U}^{\times}MX∣U=OU×, and let X = X _ X ∂ X = X _ ∖ X ∘ del X=X_\\X^(@)\partial X=\underline{X} \backslash X^{\circ}∂X=X_∖X∘.
Definition 5.5. 1. For X , Y Sm k X , Y ∈ Sm k X,Y inSm_(k)X, Y \in \operatorname{Sm}_{k}X,Y∈Smk, the group lCor k ( X , Y ) lCor k ⁡ ( X , Y ) lCor_(k)(X,Y)\operatorname{lCor}_{k}(X, Y)lCork⁡(X,Y) consisting of finite log correspondences from X X XXX to Y Y YYY is the free abelian group on integral closed subschemes Z _ X _ × Y _ Z _ ⊂ X _ × Y _ Z_subX_xxY_\underline{Z} \subset \underline{X} \times \underline{Y}Z_⊂X_×Y_ such that
(i) Z _ X _ Z _ → X _ Z_rarrX_\underline{Z} \rightarrow \underline{X}Z_→X_ is finite and is surjective to a component of X _ X _ X_\underline{X}X_.
(ii) Let Z N Z N Z^(N)Z^{N}ZN be the log scheme with underlying scheme the normalization ν : Z _ N ν : Z _ N → nu:Z_^(N)rarr\nu: \underline{Z}^{N} \rightarrowν:Z_N→ X × Y X × Y X xx YX \times YX×Y of Z _ Z _ Z_\underline{Z}Z_ and log log log\loglog structure ( ν p 1 ) log M X O Z N ν ∘ p 1 log ∗ M X → O Z N (nu@p_(1))_(log)^(**)M_(X)rarrO_(Z^(N))\left(\nu \circ p_{1}\right)_{\log }^{*} \mathcal{M}_{X} \rightarrow \mathcal{O}_{Z^{N}}(ν∘p1)log∗MX→OZN. Here M X O X M X → O X M_(X)rarrO_(X)\mathcal{M}_{X} \rightarrow \mathcal{O}_{X}MX→OX is the given log structure on X X XXX and ( ν p 1 ) log M X O Z ν ∘ p 1 log ∗ M X → O Z (nu@p_(1))_(log)^(**)M_(X)rarrO_(Z)\left(\nu \circ p_{1}\right)_{\log }^{*} \mathcal{M}_{X} \rightarrow \mathcal{O}_{Z}(ν∘p1)log∗MX→OZ is the log structure induced by the pre-log structure ( ν p 1 ) 1 M X ( ν p 1 ) 1 O X O Z ν ∘ p 1 − 1 M X → ν ∘ p 1 − 1 O X → O Z (nu@p_(1))^(-1)M_(X)rarr(nu@p_(1))^(-1)O_(X)rarrO_(Z)\left(\nu \circ p_{1}\right)^{-1} \mathcal{M}_{X} \rightarrow\left(\nu \circ p_{1}\right)^{-1} \mathcal{O}_{X} \rightarrow \mathcal{O}_{Z}(ν∘p1)−1MX→(ν∘p1)−1OX→OZ. Then the map of schemes p 2 ν : Z _ N Y _ p 2 ∘ ν : Z _ N → Y _ p_(2)@nu:Z_^(N)rarrY_p_{2} \circ \nu: \underline{Z}^{N} \rightarrow \underline{Y}p2∘ν:Z_N→Y_ extends to a map of log log log\loglog schemes Z N Y Z N → Y Z^(N)rarr YZ^{N} \rightarrow YZN→Y.
Remark 5.6. It follows from (i) and (ii) above that, for Z _ lor k ( X , Y ) Z _ ∈ lor k ⁡ ( X , Y ) Z_inlor_(k)(X,Y)\underline{Z} \in \operatorname{lor}_{k}(X, Y)Z_∈lork⁡(X,Y), the restriction of Z _ Z _ Z_\underline{Z}Z_ to a cycle on the open subset X × Y X ∘ × Y ∘ X^(@)xxY^(@)X^{\circ} \times Y^{\circ}X∘×Y∘ of X _ × Y _ X _ × Y _ X_xxY_\underline{X} \times \underline{Y}X_×Y_ actually lands in Cor k ( X , Y ) Cor k ⁡ X ∘ , Y ∘ Cor_(k)(X^(@),Y^(@))\operatorname{Cor}_{k}\left(X^{\circ}, Y^{\circ}\right)Cork⁡(X∘,Y∘). Moreover, by [19, LemMA 2.3.1], if the extension in (ii) exists, it is unique, so there is no need to include this as part of the data. In particular, the restriction map lor k ( X , Y ) Cor k ( X , Y ) lor k ⁡ ( X , Y ) → Cor k ⁡ X ∘ , Y ∘ lor_(k)(X,Y)rarrCor_(k)(X^(@),Y^(@))\operatorname{lor}_{k}(X, Y) \rightarrow \operatorname{Cor}_{k}\left(X^{\circ}, Y^{\circ}\right)lork⁡(X,Y)→Cork⁡(X∘,Y∘) is injective ([19, LEMMA 2.3.2]).
The condition that there exists a map of log log log\loglog schemes ( Z N , ( p 1 ν ) log M X ) Z N , p 1 ∘ ν log ∗ M X → (Z^(N),(p_(1)@nu)_(log)^(**)M_(X))rarr\left(Z^{N},\left(p_{1} \circ \nu\right)_{\log }^{*} \mathcal{M}_{X}\right) \rightarrow(ZN,(p1∘ν)log∗MX)→ ( Y , M Y ) Y , M Y (Y,M_(Y))\left(Y, \mathcal{M}_{Y}\right)(Y,MY) extending p 2 v : Z _ N Y _ p 2 ∘ v : Z _ N → Y _ p_(2)@v:Z_^(N)rarrY_p_{2} \circ v: \underline{Z}^{N} \rightarrow \underline{Y}p2∘v:Z_N→Y_ is analogous to the modulus condition
v ( D × Y ) v ( X × E ) v ∗ ( D × Y ) ≥ v ∗ ( X × E ) v^(**)(D xx Y) >= v^(**)(X xx E)v^{*}(D \times Y) \geq v^{*}(X \times E)v∗(D×Y)≥v∗(X×E)
for a subvariety W X D × Y E W ⊂ X ∖ D × Y ∖ E W sub X\\D xx Y\\EW \subset X \backslash D \times Y \backslash EW⊂X∖D×Y∖E to define a finite correspondence of modulus pairs from ( X , D ) ( X , D ) (X,D)(X, D)(X,D) to ( Y , E ) ( Y , E ) (Y,E)(Y, E)(Y,E).
For the composition law, the proof of [19, LEMMA 2.3.3] shows that, given elementary log log log\loglog correspondences W _ lor k ( X , Y ) W _ ∈ lor k ⁡ ( X , Y ) W_inlor_(k)(X,Y)\underline{W} \in \operatorname{lor}_{k}(X, Y)W_∈lork⁡(X,Y), and W _ 1 Cor k ( Y , Z ) W _ ′ ∈ 1 Cor k ⁡ ( Y , Z ) W_^(')in1Cor_(k)(Y,Z)\underline{W}^{\prime} \in 1 \operatorname{Cor}_{k}(Y, Z)W_′∈1Cork⁡(Y,Z), each integral component R _ R _ R_\underline{R}R_ of W _ × Z _ X _ × W _ W _ × Z _ ∩ X _ × W ′ _ W_xxZ_nnX_xxW^(')_\underline{W} \times \underline{Z} \cap \underline{X} \times \underline{W^{\prime}}W_×Z_∩X_×W′_ is the underlying scheme of a (unique!) elementary log correspondence R _ lor k ( X , Z ) R _ ∈ lor k ⁡ ( X , Z ) R_inlor_(k)(X,Z)\underline{R} \in \operatorname{lor}_{k}(X, Z)R_∈lork⁡(X,Z). It is then easy to show that there is a unique composition law
: lCor k ( Y , Z ) × lor k ( X , Y ) lor k ( X , Z ) ∘ : lCor k ⁡ ( Y , Z ) × lor k ⁡ ( X , Y ) → lor k ⁡ ( X , Z ) @:lCor_(k)(Y,Z)xxlor_(k)(X,Y)rarrlor_(k)(X,Z)\circ: \operatorname{lCor}_{k}(Y, Z) \times \operatorname{lor}_{k}(X, Y) \rightarrow \operatorname{lor}_{k}(X, Z)∘:lCork⁡(Y,Z)×lork⁡(X,Y)→lork⁡(X,Z)
that is compatible with the composition law in C o r k C o r k Cor_(k)\mathrm{Cor}_{k}Cork via the respective restriction maps.
This defines the additive category of finite log correspondences 1 C o r k 1 C o r k 1Cor_(k)\mathrm{1Cor}_{k}1Cork with the same objects as for 1 S m k 1 S m k 1Sm_(k)1 \mathrm{Sm}_{k}1Smk, giving the category of presheaves with log transfers, P S S T k P S S T k PSST_(k)\mathrm{PSST}_{k}PSSTk, defined as the category of additive presheaves of abelian groups on 1 lor k 1 lor k 1lor_(k)1 \operatorname{lor}_{k}1lork. For a log scheme X 1 S m k X ∈ 1 S m k X in1Sm_(k)X \in 1 \mathrm{Sm}_{k}X∈1Smk, let Z l t r ( X ) Z l t r ( X ) Z_(ltr)(X)\mathbb{Z}_{\mathrm{ltr}}(X)Zltr(X) denote the representable presheaf
Z 1 t r ( X ) ( Y ) := lor k ( Y , X ) Z 1 t r ( X ) ( Y ) := lor k ⁡ ( Y , X ) Z_(1tr)(X)(Y):=lor_(k)(Y,X)\mathbb{Z}_{1 \mathrm{tr}}(X)(Y):=\operatorname{lor}_{k}(Y, X)Z1tr(X)(Y):=lork⁡(Y,X)
The fiber product of log schemes induces a tensor product structure on lPST k lPST k lPST_(k)\operatorname{lPST}_{k}lPSTk.
The next step is to define the log version of the Nisnevich topology.
A morphism of log schemes f : ( X , M X O X ) , ( Y , M Y O Y ) f : X , M X → O X , Y , M Y → O Y f:(X,M_(X)rarrO_(X)),(Y,M_(Y)rarrO_(Y))f:\left(X, \mathcal{M}_{X} \rightarrow \mathcal{O}_{X}\right),\left(Y, \mathcal{M}_{Y} \rightarrow \mathcal{O}_{Y}\right)f:(X,MX→OX),(Y,MY→OY) is strict if the map of log log log\loglog structures f M Y M X f ∗ M Y → M X f^(**)M_(Y)rarrM_(X)f^{*} \mathcal{M}_{Y} \rightarrow \mathcal{M}_{X}f∗MY→MX is an isomorphism. An elementary log Nisnevich square is a cartesian square in the category of fs log schemes
where f f fff is strict étale, g g ggg is an open immersion, and f f fff induces an isomorphism on reduced schemes Y _ V _ X _ U _ Y _ ∖ V _ → X _ ∖ U _ Y_\\V_rarrX_\\U_\underline{Y} \backslash \underline{V} \rightarrow \underline{X} \backslash \underline{U}Y_∖V_→X_∖U_.
A log modification is a generalization of the notion of a log blow-up, which in turn is a morphism of log schemes modeled on the birational morphism of toric varieties given by a subdivision of the fan defining the target. We refer the reader to [19, APPENDIX A] for details. The Grothendieck topology generated by the log modifications and strict Nisnevich elementary squares is called the dividing Nisnevich topology on fs log schemes. In a sense, this is a log version of the cdh topology, where all the modifications are taking place in the boundary.
With this topology in hand, we have the subcategory lNST k lNST k lNST_(k)\operatorname{lNST}_{k}lNSTk of IPST k IPST k IPST_(k)\operatorname{IPST}_{k}IPSTk of Nisnevich sheaves with log transfers, just as for N S T k P S T k N S T k ⊂ P S T k NST_(k)subPST_(k)\mathrm{NST}_{k} \subset \mathrm{PST}_{k}NSTk⊂PSTk, by requiring that a presheaf with log transfers be a sheaf for the dividing Nisnevich topology when restricted to S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk.
Finally, we need a suitable interval object to define a good notion of homotopy invariance. This is just as for the category MDM eff ( k ) MDM eff  ⁡ ( k ) MDM^("eff ")(k)\operatorname{MDM}^{\text {eff }}(k)MDMeff ⁡(k), where we consider ¯ â—» ¯ bar(â—»)\bar{\square}◻¯ as the scheme P 1 P 1 P^(1)\mathbb{P}^{1}P1 with compactifying log structure for ( P 1 , { 1 } ) P 1 , { 1 } (P^(1),{1})\left(\mathbb{P}^{1},\{1\}\right)(P1,{1}). The product log scheme ¯ 2 â—» ¯ 2 bar(â—»)^(2)\bar{\square}^{2}◻¯2 also has the compactifying log structure for the divisor 1 × P 1 + P 1 × 1 1 × P 1 + P 1 × 1 1xxP^(1)+P^(1)xx11 \times \mathbb{P}^{1}+\mathbb{P}^{1} \times 11×P1+P1×1. However, the closure Γ ¯ m Γ ¯ m bar(Gamma)_(m)\bar{\Gamma}_{m}Γ¯m of the graph of the multiplication map m : ¯ 2 ¯ m : â—» ¯ 2 → â—» ¯ m: bar(â—»)^(2)rarr bar(â—»)m: \bar{\square}^{2} \rightarrow \bar{\square}m:◻¯2→◻¯ is not a morphism m ~ m ~ tilde(m)\tilde{m}m~ in 1 Cor k 1 Cor k 1Cor_(k)1 \operatorname{Cor}_{k}1Cork, as the requirement that the map of Γ ¯ m Γ ¯ m bar(Gamma)_(m)\bar{\Gamma}_{m}Γ¯m to ¯ 2 â—» ¯ 2 bar(â—»)^(2)\bar{\square}^{2}◻¯2 be finite is not satisfied.
Another way to look at this is to note that the projection Γ ¯ m ¯ 2 Γ ¯ m → â—» ¯ 2 bar(Gamma)_(m)rarr bar(â—»)^(2)\bar{\Gamma}_{m} \rightarrow \bar{\square}^{2}Γ¯m→◻¯2 is a cover of ¯ 2 â—» ¯ 2 bar(â—»)^(2)\bar{\square}^{2}◻¯2 in the dividing Nisnevich topology, and becomes an isomorphism after d d ddd Nis-localization. In a sense, this allows one to consider the sheaf a d N i s ¯ a d N i s â—» ¯ a_(dNis) bar(â—»)a_{\mathrm{dNis}} \bar{\square}adNis◻¯ as a version of a cylinder object and allows many of the constructions of Morel-Voevodsky for a site with interval to go through, although there are occasional technical difficulties that arise.
Definition 5.7. The tensor triangulated category of effective log motives over k , logDM e f f ( k ) k , logDM ⁡ e f f ( k ) k,logDM ^(eff)(k)k, \operatorname{logDM}{ }^{\mathrm{eff}}(k)k,logDM⁡eff(k), is the Verdier localization of the derived category D ( P S S T k ) D P S S T k D(PSST_(k))D\left(\mathrm{PSST}_{k}\right)D(PSSTk) with respect to the localizing subcategory generated by:
(IMV) for an elementary log Nisnevich square
we have the complex
Z l t r ( V ) Z l t r ( U ) Z l t r ( Y ) Z l t r ( X ) Z l t r ( V ) → Z l t r ( U ) ⊕ Z l t r ( Y ) → Z l t r ( X ) Z_(ltr)(V)rarrZ_(ltr)(U)o+Z_(ltr)(Y)rarrZ_(ltr)(X)\mathbb{Z}_{\mathrm{ltr}}(V) \rightarrow \mathbb{Z}_{\mathrm{ltr}}(U) \oplus \mathbb{Z}_{\mathrm{ltr}}(Y) \rightarrow \mathbb{Z}_{\mathrm{ltr}}(X)Zltr(V)→Zltr(U)⊕Zltr(Y)→Zltr(X)
(IM) For a log modification f : Y X f : Y → X f:Y rarr Xf: Y \rightarrow Xf:Y→X in 1 S m k 1 S m k 1Sm_(k)1 \mathrm{Sm}_{k}1Smk, we have the complex
Z l t r ( Y ) Z l t r ( X ) Z l t r ( Y ) → Z l t r ( X ) Z_(ltr)(Y)rarrZ_(ltr)(X)\mathbb{Z}_{\mathrm{ltr}}(Y) \rightarrow \mathbb{Z}_{\mathrm{ltr}}(X)Zltr(Y)→Zltr(X)
(lCI) For X 1 S m k X ∈ 1 S m k X in1Sm_(k)X \in 1 \mathrm{Sm}_{k}X∈1Smk, we have the complex
Z l t r ( X × ) Z l t r ( X ) Z l t r ( X × ⊵ ) → Z l t r ( X ) Z_(ltr)(X xx⊵)rarrZ_(ltr)(X)\mathbb{Z}_{\mathrm{ltr}}(X \times \unrhd) \rightarrow \mathbb{Z}_{\mathrm{ltr}}(X)Zltr(X×⊵)→Zltr(X)
For each fs smooth log scheme X 1 Sm k X ∈ 1 Sm k X in1Sm_(k)X \in 1 \operatorname{Sm}_{k}X∈1Smk, the image of Z ltr ( X ) Z ltr  ( X ) Z_("ltr ")(X)\mathbb{Z}_{\text {ltr }}(X)Zltr (X) in logDM eff ( k ) logDM ⁡ eff  ( k ) logDM ^("eff ")(k)\operatorname{logDM}{ }^{\text {eff }}(k)logDM⁡eff (k) is the effective log motive l M eff ( X ) l M eff  ( X ) lM^("eff ")(X)\mathrm{lM}^{\text {eff }}(X)lMeff (X), giving the functor
l M e f f : 1 S m k logDM e f f ( k ) l M e f f : 1 S m k → logDM ⁡ e f f ( k ) lM^(eff):1Sm_(k)rarr logDM ^(eff)(k)\mathrm{lM}^{\mathrm{eff}}: 1 \mathrm{Sm}_{k} \rightarrow \operatorname{logDM}{ }^{\mathrm{eff}}(k)lMeff:1Smk→logDM⁡eff(k)
The functor I M eff I M eff  IM^("eff ")\mathrm{IM}^{\text {eff }}IMeff  shares many of the formal properties of M eff : Sm k D M eff ( k ) M eff  : Sm k → D M eff  ( k ) M^("eff "):Sm_(k)rarrDM^("eff ")(k)M^{\text {eff }}: \operatorname{Sm}_{k} \rightarrow \mathrm{DM}^{\text {eff }}(k)Meff :Smk→DMeff (k); we refer the reader to the [19, INTRoDUction] for an overview.
Questions of representing known constructions such as the higher Chow groups with modulus in log D M eff ( k ) log ⁡ D M eff  ( k ) log DM^("eff ")(k)\log \mathrm{DM}^{\text {eff }}(k)log⁡DMeff (k), or finding direct connections of log D M eff ( k ) log ⁡ D M eff  ( k ) log DM^("eff ")(k)\log \mathrm{DM}^{\text {eff }}(k)log⁡DMeff (k) with the category MDM eff ( k ) MDM eff  ⁡ ( k ) MDM^("eff ")(k)\operatorname{MDM}^{\text {eff }}(k)MDMeff ⁡(k) are not discussed in [19]. However, for ( X , D ) ( X , D ) (X,D)(X, D)(X,D) a proper modulus pair, one has the log scheme l ( X , D ) l ( X , D ) l(X,D)l(X, D)l(X,D), defined using the Deligne-Faltings log structure on X X XXX associated to the ideal sheaf O X ( D ) O X ( − D ) O_(X)(-D)\mathcal{O}_{X}(-D)OX(−D). In general, this is not saturated. Still, there should be presheaves with log log log\loglog transfers Z l t r ( X , D ) Z l t r ( X , D ) Z_(ltr)(X,D)\mathbb{Z}_{\mathrm{ltr}}(X, D)Zltr(X,D) and Z l t r c ( X , D ) Z l t r c ( X , D ) Z_(ltr)^(c)(X,D)\mathbb{Z}_{\mathrm{ltr}}^{c}(X, D)Zltrc(X,D) using finite and quasi-finite "log correspondences," with value on Y 1 Sm k Y ∈ 1 Sm k Y in1Sm_(k)Y \in 1 \operatorname{Sm}_{k}Y∈1Smk the free abelian group of integral subschemes W W WWW of Y _ × X Y _ × X Y_xx X\underline{Y} \times XY_×X that admit a map of log schemes ( W N , ( p 1 ν ) ( M Y ) ) l ( X , D ) W N , p 1 ∘ ν ∗ M Y → l ( X , D ) (W^(N),(p_(1)@nu)^(**)(M_(Y)))rarr l(X,D)\left(W^{N},\left(p_{1} \circ \nu\right)^{*}\left(\mathcal{M}_{Y}\right)\right) \rightarrow l(X, D)(WN,(p1∘ν)∗(MY))→l(X,D), as in the definition of 1 Cor k ( , ) 1 Cor k ⁡ ( − , − ) 1Cor_(k)(-,-)1 \operatorname{Cor}_{k}(-,-)1Cork⁡(−,−). One could also expect to have presheaves l z ( X , D , r ) l z ( X , D , r ) lz(X,D,r)l z(X, D, r)lz(X,D,r) similarly defined, and corresponding to the presheaves with modulus transfers z ( X , D , r ) z ( X , D , r ) z(X,D,r)z(X, D, r)z(X,D,r) constructed by Miyazaki. These could be used to give a map
H p ( z r equi ( X , D , ) ) Hom log D M eff ( k ) ( Z ( 0 ) [ p ] , M ( l z ( X , D , r ) ) ) H p z r equi  ( X , D , ∗ ) → Hom log ⁡ D M eff  ( k ) ⁡ ( Z ( 0 ) [ p ] , M ( l z ( X , D , r ) ) ) H_(p)(z_(r)^("equi ")(X,D,**))rarrHom_(log DM^("eff ")(k))(Z(0)[p],M(lz(X,D,r)))H_{p}\left(z_{r}^{\text {equi }}(X, D, *)\right) \rightarrow \operatorname{Hom}_{\log D M^{\text {eff }}(k)}(\mathbb{Z}(0)[p], M(l z(X, D, r)))Hp(zrequi (X,D,∗))→Homlog⁡DMeff (k)⁡(Z(0)[p],M(lz(X,D,r)))
We have briefly mentioned reciprocity sheaves in our discussion of motives with modulus. There is a nice connection of logDM eff ( k ) logDM ⁡ eff  ( k ) logDM ^("eff ")(k)\operatorname{logDM}{ }^{\text {eff }}(k)logDM⁡eff (k) with the theory of reciprocity sheaves, so we take the opportunity to say a few words about reciprocity sheaves before we describe the theorem of Shuji Saito, which gives the connection between these two theories.
The notion of a reciprocity sheaf and its relation to motives with modulus goes back to the theorem of Rosenlicht-Serre. In our discussion of reciprocity sheaves, we work over a fixed perfect field k k kkk.
Theorem 5.8 (Rosenlicht-Serre [ 109 [ 109 [109[109[109, III]). Let k k kkk be a perfect field, let C C CCC be a smooth complete curve over k k kkk, let G G GGG be an smooth commutative algebraic group over k k kkk, and let f f fff : C G C → G C rarr GC \rightarrow GC→G be a rational map over k k kkk. Let S C S ⊂ C S sub CS \subset CS⊂C be a finite subset such that f f fff is a morphism on C S C ∖ S C\\SC \backslash SC∖S. Then there is an effective divisor D D DDD supported in S S SSS such that, for g g ggg a rational function on C C CCC with g 1 mod D g ≡ 1 mod D g-=1mod Dg \equiv 1 \bmod Dg≡1modD, one has
P C S ord P ( g ) f ( P ) = 0 ∑ P ∈ C ∖ S   ord P ⁡ ( g ) ⋅ f ( P ) = 0 sum_(P in C\\S)ord_(P)(g)*f(P)=0\sum_{P \in C \backslash S} \operatorname{ord}_{P}(g) \cdot f(P)=0∑P∈C∖SordP⁡(g)⋅f(P)=0
in G G GGG.
In [72], reciprocity functors and reciprocity sheaves are defined. We will just give a sketch. One first defines for F F FFF a presheaf with transfers (in the Voevodsky sense), and for a proper modulus pair ( X , D ) ( X , D ) (X,D)(X, D)(X,D) with a section a F ( X D ) a ∈ F ( X ∖ D ) a in F(X\\D)a \in F(X \backslash D)a∈F(X∖D), what it means for a a aaa to have modulus D D DDD. As an example, if p : C X p : C → X p:C rarr Xp: C \rightarrow Xp:C→X is a non-constant morphism of a smooth proper integral curve C C CCC over k k kkk to X X XXX with p ( C ) p ( C ) p(C)p(C)p(C) not contained in D D DDD, and g g ggg is a rational function on C C CCC such that g 1 mod p ( D ) g ≡ 1 mod p ∗ ( D ) g-=1modp^(**)(D)g \equiv 1 \bmod p^{*}(D)g≡1modp∗(D), then one is required to have
a ( p ( div ( g ) ) ) = 0 F ( Spec k ) a p ∗ ( div ⁡ ( g ) ) = 0 ∈ F ( Spec ⁡ k ) a(p_(**)(div(g)))=0in F(Spec k)a\left(p_{*}(\operatorname{div}(g))\right)=0 \in F(\operatorname{Spec} k)a(p∗(div⁡(g)))=0∈F(Spec⁡k)
Here, for a 0 -cycle x n x x ∑ x   n x â‹… x sum_(x)n_(x)*x\sum_{x} n_{x} \cdot x∑xnxâ‹…x on X D , a ( x n x x ) = x n x p x i x ( a ) F ( Spec k ) X ∖ D , a ∑ x   n x â‹… x = ∑ x   n x â‹… p x ∗ i x ∗ ( a ) ∈ F ( Spec ⁡ k ) X\\D,a(sum_(x)n_(x)*x)=sum_(x)n_(x)*p_(x**i_(x)^(**))(a)in F(Spec k)X \backslash D, a\left(\sum_{x} n_{x} \cdot x\right)=\sum_{x} n_{x} \cdot p_{x * i_{x}^{*}}(a) \in F(\operatorname{Spec} k)X∖D,a(∑xnxâ‹…x)=∑xnxâ‹…px∗ix∗(a)∈F(Spec⁡k), where for a closed point x x xxx of X D , i x : x X D X ∖ D , i x : x → X ∖ D X\\D,i_(x):x rarr X\\DX \backslash D, i_{x}: x \rightarrow X \backslash DX∖D,ix:x→X∖D is the inclusion and p x : x Spec k p x : x → Spec ⁡ k p_(x):x rarr Spec kp_{x}: x \rightarrow \operatorname{Spec} kpx:x→Spec⁡k is the (finite) structure morphism. In general, one imposes a similar condition in F ( S ) F ( S ) F(S)F(S)F(S) for a "relative curve" on X × S X × S X xx SX \times SX×S over some smooth base scheme S S SSS.
A presheaf with transfers F F FFF is a reciprocity sheaf if for each quasi-affine U U UUU and section a F ( U ) a ∈ F ( U ) a in F(U)a \in F(U)a∈F(U), there is a proper modulus pair ( X , D ) ( X , D ) (X,D)(X, D)(X,D) with U = X D U = X ∖ D U=X\\DU=X \backslash DU=X∖D such that a a aaa has modulus D D DDD. Roughly speaking, one should think that each section of F F FFF has "bounded ramification," although the "ramification" for F F FFF itself may be unbounded.
This definition is not quite accurate, as a slightly different notion of "modulus pair" from what we have defined here is used in [72]. A more elegant definition of reciprocity sheaf is given in [73]. This new notion is a bit more restrictive than the old one, but by [73, THEOREM 2], the two notions agree on for F M N S T k _ F ∈ M N S T k _ F inMNST_(k)_F \in \underline{\mathrm{MNST}_{k}}F∈MNSTk_.
Using the definition of [73], the reciprocity sheaves define a the full subcategory R S T k R S T k RST_(k)\mathbf{R S T}_{k}RSTk of P S T k P S T k PST_(k)\mathrm{PST}_{k}PSTk, strictly containing the subcategory H I k P S T k H I k ⊂ P S T k HI_(k)subPST_(k)\mathbf{H I}_{k} \subset \mathrm{PST}_{k}HIk⊂PSTk of A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy invariant presheaves with transfer. There is also the subcategory R S T N i s , k R S T N i s , k RST_(Nis,k)\mathbf{R S T}_{\mathrm{Nis}, k}RSTNis,k of N S T k N S T k NST_(k)\mathrm{NST}_{k}NSTk, consisting of those reciprocity presheaves that are Nisnevich sheaves.
Some examples of non-homotopy invariant sheaves in R S T N i s , k R S T N i s , k RST_(Nis,k)\mathbf{R S T}_{\mathrm{Nis}, k}RSTNis,k include the sheaf of n n nnn-forms over k , X Ω X / k n k , X ↦ Ω X / k n k,X|->Omega_(X//k)^(n)k, X \mapsto \Omega_{X / k}^{n}k,X↦ΩX/kn, the sheaf of absolute n n nnn-forms, X Ω X n X ↦ Ω X n X|->Omega_(X)^(n)X \mapsto \Omega_{X}^{n}X↦ΩXn, and for k k kkk of positive characteristic, the truncated de Rham-Witt sheaves, X W m Ω X n X ↦ W m Ω X n X|->W_(m)Omega_(X)^(n)X \mapsto W_{m} \Omega_{X}^{n}X↦WmΩXn. The representable sheaf of a commutative algebraic group G G GGG over k , X G ( X ) k , X ↦ G ( X ) k,X|->G(X)k, X \mapsto G(X)k,X↦G(X), is in R S T Nis , k R S T Nis  , k RST_("Nis ",k)\mathbf{R S T}_{\text {Nis }, k}RSTNis ,k, and for some G G GGG (e.g. G = G m n G = G m n G=G_(m)^(n)G=\mathbb{G}_{m}^{n}G=Gmn ) this is also A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy invariant. This is not the case for unipotent G G GGG (e.g. G = G a n ) G = G a n {:G=G_(a)^(n))\left.G=\mathbb{G}_{a}^{n}\right)G=Gan).
Here is the promised theorem of Saito. For a sheaf G NNST k G ∈ NNST k G inNNST_(k)G \in \operatorname{NNST}_{k}G∈NNSTk, we say that G G GGG is
H d N i s ( X , G X d N i s ) H d N i s ( X × ¯ , G X × ¯ d N i s ) H d N i s ∗ X , G ∣ X d N i s → H d N i s ∗ X × â—» ¯ , G ∣ X × â—» ¯ d N i s H_(dNis)^(**)(X,G_(∣X_(dNis)))rarrH_(dNis)^(**)(X xx( bar(â—»)),G_(∣X xx bar(â—»)_(dNis)))H_{d \mathrm{Nis}}^{*}\left(X, G_{\mid X_{d \mathrm{Nis}}}\right) \rightarrow H_{d \mathrm{Nis}}^{*}\left(X \times \bar{\square}, G_{\mid X \times \bar{\square}_{d \mathrm{Nis}}}\right)HdNis∗(X,G∣XdNis)→HdNis∗(X×◻¯,G∣X×◻¯dNis)
induced by the projection X × ¯ X X × â—» ¯ → X X xx bar(â—»)rarr XX \times \bar{\square} \rightarrow XX×◻¯→X is an isomorphism. Here d d ddd Nis refers to the divided Nisnevich site.

Theorem 5.9 (Saito [105, THEOREM 0.2]). There exists a fully faithful exact functor

log : R S T N i s , k lNST k log : R S T N i s , k → lNST k log:RST_(Nis,k)rarrlNST_(k)\log : \mathbf{R S T}_{\mathrm{Nis}, k} \rightarrow \operatorname{lNST}_{k}log:RSTNis,k→lNSTk
such that log ( F ) log ⁡ ( F ) log(F)\log (F)log⁡(F) is strictly ¯ â—» ¯ bar(â—»)\bar{\square}◻¯-invariantfor every F R S T N i s , k F ∈ R S T N i s , k F inRST_(Nis,k)F \in \mathbf{R S T}_{\mathrm{Nis}, k}F∈RSTNis,k. Moreover, for each X Sm k X ∈ Sm k X inSm_(k)X \in \operatorname{Sm}_{k}X∈Smk, there is a natural isomorphism
H N i s i ( X , F X ) Hom logDM e f f ( k ) ( lM e f f ( X ) , log ( F ) [ i ] ) H N i s i X , F ∣ X ≅ Hom logDM e f f ⁡ ( k ) ⁡ lM e f f ⁡ ( X ) , log ⁡ ( F ) [ i ] H_(Nis)^(i)(X,F_(∣X))~=Hom_(logDM^(eff)(k))(lM^(eff)(X),log(F)[i])H_{\mathrm{Nis}}^{i}\left(X, F_{\mid X}\right) \cong \operatorname{Hom}_{\operatorname{logDM}^{\mathrm{eff}}(k)}\left(\operatorname{lM}^{\mathrm{eff}}(X), \log (F)[i]\right)HNisi(X,F∣X)≅HomlogDMeff⁡(k)⁡(lMeff⁡(X),log⁡(F)[i])

6. p p ppp-ADIC ÉTALE MOTIVIC COHOMOLOGY AND p p ppp-ADIC HODGE
THEORY

We discuss yet another theory of motivic cohomology that is not A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy invariant.
Working over a base-field k k kkk and for m m mmm prime to the characteristic, we have the isomorphism of the étale sheafification Z / m ( r ) ét Z / m ( r ) ét  Z//m(r)_("ét ")\mathbb{Z} / m(r)_{\text {ét }}Z/m(r)ét  with the étale sheaf μ m r μ m ⊗ r mu_(m)^(ox r)\mu_{m}^{\otimes r}μm⊗r. The étale sheaves Z / m ( r ) ét Z / m ( r ) ét  Z//m(r)_("ét ")\mathbb{Z} / m(r)_{\text {ét }}Z/m(r)ét  can be considered as objects in a version of Voevodsky's DM constructed using the étale topology rather than the Nisnevich topology, and their categorical cohomology agrees with the usual étale cohomology [127, cHAP. 5, §3.3]. In particular, the complexes Z / m ( r ) ét Z / m ( r ) ét  Z//m(r)_("ét ")\mathbb{Z} / m(r)_{\text {ét }}Z/m(r)ét  have A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy invariant étale cohomology.
On the other hand, if k k kkk has characteristic p > 0 p > 0 p > 0p>0p>0, we have the isomorphism [52] of the Nisnevich sheaves on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk,
(6.1) Z / p n ( r ) W n Ω log r [ r ] (6.1) Z / p n ( r ) ≅ W n Ω log r [ − r ] {:(6.1)Z//p^(n)(r)~=W_(n)Omega_(log)^(r)[-r]:}\begin{equation*} \mathbb{Z} / p^{n}(r) \cong W_{n} \Omega_{\log }^{r}[-r] \tag{6.1} \end{equation*}(6.1)Z/pn(r)≅WnΩlogr[−r]
hence of étale sheaves
Z / p n ( r ) é t W n Ω log r [ r ] Z / p n ( r ) é t ≅ W n Ω log r [ − r ] Z//p^(n)(r)_(ét)~=W_(n)Omega_(log)^(r)[-r]\mathbb{Z} / p^{n}(r)_{\mathrm{ét}} \cong W_{n} \Omega_{\log }^{r}[-r]Z/pn(r)ét≅WnΩlogr[−r]
Here something strange happens: the étale sheaf Z / p n ( r ) ett Z / p n ( r ) ett  Z//p^(n)(r)_("ett ")\mathbb{Z} / p^{n}(r)_{\text {ett }}Z/pn(r)ett  is no longer strictly homotopy invariant! In fact, the existence of the Artin-Schreyer étale cover A 1 A 1 A 1 → A 1 A^(1)rarrA^(1)\mathbb{A}^{1} \rightarrow \mathbb{A}^{1}A1→A1 of degree p p ppp implies that the étale version of D M eff ( k ) D M eff  ( k ) DM^("eff ")(k)\mathrm{DM}^{\text {eff }}(k)DMeff (k) with coefficients modulo p n p n p^(n)p^{n}pn is zero if k k kkk has characteristic p > 0 p > 0 p > 0p>0p>0 [127, CHAP. 5, PROPOSITION 3.3.3]. Thus Z / p n ( r ) ét Z / p n ( r ) ét  Z//p^(n)(r)_("ét ")\mathbb{Z} / p^{n}(r)_{\text {ét }}Z/pn(r)ét  leaves the world of Voevodsky's motives and motivic cohomology.
For S = Spec Λ S = Spec ⁡ Λ S=Spec LambdaS=\operatorname{Spec} \LambdaS=Spec⁡Λ, with Λ Î› Lambda\LambdaΛ a mixed characteristic ( 0 , p ) d v r ( 0 , p ) d v r (0,p)dvr(0, p) \mathrm{dvr}(0,p)dvr, the complex Z / p n ( r ) et Z / p n ( r ) et  Z//p^(n)(r)_("et ")\mathbb{Z} / p^{n}(r)_{\text {et }}Z/pn(r)et  on Sm S , ét Sm S ,  ét  Sm_(S," ét ")\operatorname{Sm}_{S, \text { ét }}SmS, ét  yields an interesting gluing of Z / p n ( r ) ét = μ p n r Z / p n ( r ) ét  = μ p n ⊗ r Z//p^(n)(r)_("ét ")=mu_(p^(n))^(ox r)\mathbb{Z} / p^{n}(r)_{\text {ét }}=\mu_{p^{n}}^{\otimes r}Z/pn(r)ét =μpn⊗r over the characteristic zero quotient field of Λ Î› Lambda\LambdaΛ and Z / p n ( r ) et = W n Ω log r [ r ] Z / p n ( r ) et  = W n Ω log r [ − r ] Z//p^(n)(r)_("et ")=W_(n)Omega_(log)^(r)[-r]\mathbb{Z} / p^{n}(r)_{\text {et }}=W_{n} \Omega_{\log }^{r}[-r]Z/pn(r)et =WnΩlogr[−r] over the characteristic p p ppp residue field. The positive characteristic part again says that we have left homotopy invariance behind.
The complexes Z / p n ( r ) ét Z / p n ( r ) ét  Z//p^(n)(r)_("ét ")\mathbb{Z} / p^{n}(r)_{\text {ét }}Z/pn(r)ét  have an interesting connection with a certain complex of sheaves arising in p p ppp-adic Hodge theory. A version of this complex first appears in the paper [40] of Fontaine-Messing, and plays an important role in the proof of their main result. Its construction was reinterpreted by Kurihara [82], relying on the work of Bloch-Kato [28] and Kato [79], and was generalized by Sato [106]. Geisser [49], following work of Schneider [108], established the connection of the Fontaine-Messing/Kurihara/Sato complex with Z / p n ( r ) êt Z / p n ( r ) êt  Z//p^(n)(r)_("êt ")\mathbb{Z} / p^{n}(r)_{\text {êt }}Z/pn(r)êt  in the case of a smooth degeneration, and this connection was partially extended by Zhong [128] to the semi-stable case.
In their recent work on integral p p ppp-adic Hodge theory, Bhatt-Morrow-Scholze [18] have introduced a "motivic filtration" on p p ppp-adic étale K K KKK-theory, relying on a Postnikov tower for topological cyclic homology, and the layers in this tower have been identified with the pro-system { Z / p n ( r ) ét } n Z / p n ( r ) ét  n {Z//p^(n)(r)_("ét ")}_(n)\left\{\mathbb{Z} / p^{n}(r)_{\text {ét }}\right\}_{n}{Z/pn(r)ét }n in a work-in-progress [16] by Bhargav Bhatt and Akhil Mathew.
Our goal in this section is to give some details of the story sketched above.
We first discuss the papers of Bloch-Kato, Fontaine-Messing, Kurihara and Sato without reference to all the advances in p p ppp-adic Hodge theory that followed these works; we wanted to give the reader just enough background to put the connections with motivic
cohomology in context. We will then describe the works of Geisser and Zhong, as well as results of Geisser-Hesselholt that form some of the foundations for the work of BhattMorrow-Scholze. We conclude with a description of the Bhatt-Morrow-Scholze motivic tower and its connection with the p p ppp-adic cycle complexes.
We refer the reader to [15] for background on crystalline cohomology.

6.1. A quick overview of some p p p\boldsymbol{p}p-adic Hodge theory

We begin with a few comments on the paper of Bloch and Kato [28], which we have already mentioned in our discussion of the Beilinson-Lichtenbaum conjectures. They consider the spectrum S S SSS of a complete dvr Λ dvr ⁡ Λ dvr Lambda\operatorname{dvr} \Lambdadvr⁡Λ with generic point η = Spec K S η = Spec ⁡ K ↪ S eta=Spec K↪S\eta=\operatorname{Spec} K \hookrightarrow Sη=Spec⁡K↪S and closed point s = Spec k S s = Spec ⁡ k ↪ S s=Spec k↪Ss=\operatorname{Spec} k \hookrightarrow Ss=Spec⁡k↪S, and a smooth and proper S S SSS-scheme X S X → S X rarr SX \rightarrow SX→S with generic fiber V := X η V := X η V:=X_(eta)V:=X_{\eta}V:=Xη and special fiber Y := X s . V ¯ , Y ¯ Y := X s . V ¯ , Y ¯ Y:=X_(s). bar(V), bar(Y)Y:=X_{s} . \bar{V}, \bar{Y}Y:=Xs.V¯,Y¯ denote V , Y V , Y V,YV, YV,Y over the respective algebraic closures K ¯ K ¯ bar(K)\bar{K}K¯ and k ¯ k ¯ bar(k)\bar{k}k¯ of K K KKK and k k kkk. Let Λ ¯ Λ ¯ bar(Lambda)\bar{\Lambda}Λ¯ be the integral closure of Λ Î› Lambda\LambdaΛ in K ¯ , S ¯ := Spec Λ ¯ K ¯ , S ¯ := Spec ⁡ Λ ¯ bar(K), bar(S):=Spec bar(Lambda)\bar{K}, \bar{S}:=\operatorname{Spec} \bar{\Lambda}K¯,S¯:=Spec⁡Λ¯, and X ¯ = X × S S ¯ X ¯ = X × S S ¯ bar(X)=Xxx_(S) bar(S)\bar{X}=X \times_{S} \bar{S}X¯=X×SS¯. Let G = Gal ( K ¯ / K ) G = Gal ⁡ ( K ¯ / K ) G=Gal( bar(K)//K)G=\operatorname{Gal}(\bar{K} / K)G=Gal⁡(K¯/K) and let C C CCC denote the completion of K ¯ K ¯ bar(K)\bar{K}K¯.
The closure Y ¯ Y ¯ bar(Y)\bar{Y}Y¯ has its crystalline cohomology H crys ( Y ¯ / W ( k ¯ ) ) H crys  ∗ ( Y ¯ / W ( k ¯ ) ) H_("crys ")^(**)( bar(Y)//W( bar(k)))H_{\text {crys }}^{*}(\bar{Y} / W(\bar{k}))Hcrys ∗(Y¯/W(k¯)) with action of Frobenius, giving the p i p i p^(i)p^{i}pi-eigenspace
H crys ( Y ¯ / W ( k ¯ ) ) ( i ) H crys ( Y ¯ / W ( k ¯ ) ) Q H crys  ∗ ( Y ¯ / W ( k ¯ ) ) ( i ) ⊂ H crys  ∗ ( Y ¯ / W ( k ¯ ) ) Q H_("crys ")^(**)( bar(Y)//W( bar(k)))^((i))subH_("crys ")^(**)( bar(Y)//W( bar(k)))_(Q)H_{\text {crys }}^{*}(\bar{Y} / W(\bar{k}))^{(i)} \subset H_{\text {crys }}^{*}(\bar{Y} / W(\bar{k}))_{\mathbb{Q}}Hcrys ∗(Y¯/W(k¯))(i)⊂Hcrys ∗(Y¯/W(k¯))Q
We say Y ¯ Y ¯ bar(Y)\bar{Y}Y¯ is ordinary if
dim W ( k ¯ ) Q H c r y s m ( Y ¯ / W ( k ¯ ) ) ( i ) = dim k ¯ H m i ( Y ¯ , Ω Y ¯ / k ¯ i ) dim W ( k ¯ ) Q ⁡ H c r y s m ( Y ¯ / W ( k ¯ ) ) ( i ) = dim k ¯ ⁡ H m − i Y ¯ , Ω Y ¯ / k ¯ i dim_(W( bar(k))_(Q))H_(crys)^(m)( bar(Y)//W( bar(k)))^((i))=dim_( bar(k))H^(m-i)(( bar(Y)),Omega_( bar(Y)// bar(k))^(i))\operatorname{dim}_{W(\bar{k})_{\mathscr{Q}}} H_{\mathrm{crys}}^{m}(\bar{Y} / W(\bar{k}))^{(i)}=\operatorname{dim}_{\bar{k}} H^{m-i}\left(\bar{Y}, \Omega_{\bar{Y} / \bar{k}}^{i}\right)dimW(k¯)Q⁡Hcrysm(Y¯/W(k¯))(i)=dimk¯⁡Hm−i(Y¯,ΩY¯/k¯i)
We have the inclusions i ¯ : Y ¯ X ¯ , j ¯ : V ¯ X ¯ i ¯ : Y ¯ → X ¯ , j ¯ : V ¯ → X ¯ bar(i): bar(Y)rarr bar(X), bar(j): bar(V)rarr bar(X)\bar{i}: \bar{Y} \rightarrow \bar{X}, \bar{j}: \bar{V} \rightarrow \bar{X}i¯:Y¯→X¯,j¯:V¯→X¯ and the spectral sequence
E 2 s , t = H e t s ( Y ¯ , i ¯ R t j ¯ ( Z / p n Z ) ) H e t s + t ( V ¯ , Z / p n Z ) E 2 s , t = H e t s Y ¯ , i ¯ ∗ R t j ¯ ∗ Z / p n Z ⇒ H e t s + t V ¯ , Z / p n Z E_(2)^(s,t)=H_(et)^(s)(( bar(Y)), bar(i)^(**)R^(t) bar(j)_(**)(Z//p^(n)Z))=>H_(et)^(s+t)(( bar(V)),Z//p^(n)Z)E_{2}^{s, t}=H_{\mathrm{et}}^{s}\left(\bar{Y}, \bar{i}^{*} R^{t} \bar{j}_{*}\left(\mathbb{Z} / p^{n} \mathbb{Z}\right)\right) \Rightarrow H_{\mathrm{et}}^{s+t}\left(\bar{V}, \mathbb{Z} / p^{n} \mathbb{Z}\right)E2s,t=Hets(Y¯,i¯∗Rtj¯∗(Z/pnZ))⇒Hets+t(V¯,Z/pnZ)
inducing a descending filtration F H e t ( V ¯ , Q p ) F ∗ H e t ∗ V ¯ , Q p F^(**)H_(et)^(**)(( bar(V)),Q_(p))F^{*} H_{\mathrm{et}}^{*}\left(\bar{V}, \mathbb{Q}_{p}\right)F∗Het∗(V¯,Qp) on H e t ( V ¯ , Q p ) H e t ∗ V ¯ , Q p H_(et)^(**)(( bar(V)),Q_(p))H_{\mathrm{et}}^{*}\left(\bar{V}, \mathbb{Q}_{p}\right)Het∗(V¯,Qp) with F 0 H q = H q F 0 H q = H q F^(0)H^(q)=H^(q)F^{0} H^{q}=H^{q}F0Hq=Hq and F q + 1 H q = 0 F q + 1 H q = 0 F^(q+1)H^(q)=0F^{q+1} H^{q}=0Fq+1Hq=0.
We have the de Rham-Witt sheaf W Ω i W Ω i WOmega^(i)W \Omega^{i}WΩi on S m k ¯ S m k ¯ Sm_( bar(k))\mathrm{Sm}_{\bar{k}}Smk¯ and the sheaf of differential forms Ω / K i Ω − / K i Omega_(-//K)^(i)\Omega_{-/ K}^{i}Ω−/Ki on S m K S m K Sm_(K)\mathrm{Sm}_{K}SmK.
Theorem 6.1 (Bloch-Kato [28, тHEORem 0.7]). Suppose that k k kkk is perfect and that Y Y YYY is ordinary. Then there are natural G G GGG-equivariant isomorphisms
(i) gr q i H e t q ( V ¯ , Q p ) H c r y s q ( Y ¯ / W ( k ¯ ) ) Q ( i ) ( i ) gr q − i ⁡ H e t q V ¯ , Q p ≅ H c r y s q ( Y ¯ / W ( k ¯ ) ) Q ( i ) ( − i ) gr^(q-i)H_(et)^(q)(( bar(V)),Q_(p))~=H_(crys)^(q)( bar(Y)//W( bar(k)))_(Q)^((i))(-i)\operatorname{gr}^{q-i} H_{e t}^{q}\left(\bar{V}, \mathbb{Q}_{p}\right) \cong H_{\mathrm{crys}}^{q}(\bar{Y} / W(\bar{k}))_{\mathbb{Q}}^{(i)}(-i)grq−i⁡Hetq(V¯,Qp)≅Hcrysq(Y¯/W(k¯))Q(i)(−i),
(ii) gr q i H e t q ( V ¯ , Q p ) Z p W ( k ¯ ) H crys q ( Y ¯ , W Ω i ) Q ( i ) gr q − i ⁡ H e t q V ¯ , Q p ⊗ Z p W ( k ¯ ) ≅ H crys  q Y ¯ , W Ω i Q ( − i ) gr^(q-i)H_(et)^(q)(( bar(V)),Q_(p))ox_(Z_(p))W( bar(k))~=H_("crys ")^(q)(( bar(Y)),WOmega^(i))_(Q)(-i)\operatorname{gr}^{q-i} H_{e t}^{q}\left(\bar{V}, \mathbb{Q}_{p}\right) \otimes_{\mathbb{Z}_{p}} W(\bar{k}) \cong H_{\text {crys }}^{q}\left(\bar{Y}, W \Omega^{i}\right)_{\mathbb{Q}}(-i)grq−i⁡Hetq(V¯,Qp)⊗ZpW(k¯)≅Hcrys q(Y¯,WΩi)Q(−i),
(iii) g r q i H e t q ( V ¯ , Q p ) Q p C H q ( V , Ω V / K i ) K C ( i ) g r q − i H e t q V ¯ , Q p ⊗ Q p C ≅ H q V , Ω V / K i ⊗ K C ( − i ) gr^(q-i)H_(et)^(q)(( bar(V)),Q_(p))ox_(Q_(p))C~=H^(q)(V,Omega_(V//K)^(i))ox_(K)C(-i)\mathrm{gr}^{q-i} H_{e t}^{q}\left(\bar{V}, \mathbb{Q}_{p}\right) \otimes_{\mathbb{Q}_{p}} C \cong H^{q}\left(V, \Omega_{V / K}^{i}\right) \otimes_{K} C(-i)grq−iHetq(V¯,Qp)⊗QpC≅Hq(V,ΩV/Ki)⊗KC(−i).
We will not give any details of the proof here, but do want to mention that what ties these different theories together is the sheaf of Milnor K K KKK-groups K q M K q M K_(q)^(M)\mathcal{K}_{q}^{M}KqM. This maps to étale cohomology by the Galois symbol
θ q , m : K q M / m H e t q ( μ m q ) θ q , m : K q M / m → H e t q μ m ⊗ q theta_(q,m):K_(q)^(M)//m rarrH_(et)^(q)(mu_(m)^(ox q))\theta_{q, m}: \mathcal{K}_{q}^{M} / m \rightarrow \mathscr{H}_{\mathrm{et}}^{q}\left(\mu_{m}^{\otimes q}\right)θq,m:KqM/m→Hetq(μm⊗q)
for m m mmm prime to the characteristic, to the de Rham-Witt sheaf by the d log d log d logd \logdlog map on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk,
d log q , p n W : K q M / p n W n Ω q d log q , p n W : K q M / p n → W n Ω q dlog_(q,p^(n))^(W):K_(q)^(M)//p^(n)rarrW_(n)Omega^(q)d \log _{q, p^{n}}^{W}: \mathcal{K}_{q}^{M} / p^{n} \rightarrow W_{n} \Omega^{q}dlogq,pnW:KqM/pn→WnΩq
and to the sheaf of differential forms, by the d log d log d logd \logdlog map on S m S S m S Sm_(S)\mathrm{Sm}_{S}SmS,
d log q , p n : K q M / p n Ω / S q / p n d log q , p n : K q M / p n → Ω − / S q / p n dlog_(q,p^(n)):K_(q)^(M)//p^(n)rarrOmega_(-//S)^(q)//p^(n)d \log _{q, p^{n}}: \mathcal{K}_{q}^{M} / p^{n} \rightarrow \Omega_{-/ S}^{q} / p^{n}dlogq,pn:KqM/pn→Ω−/Sq/pn
The main structural results that underpin the proof of the Bloch-Kato theorem are two comparison isomorphisms on the sheaf level. For the first, let W n Ω log q W n Ω q W n Ω log q ⊂ W n Ω q W_(n)Omega_(log)^(q)subW_(n)Omega^(q)W_{n} \Omega_{\log }^{q} \subset W_{n} \Omega^{q}WnΩlogq⊂WnΩq be the étale subsheaf locally generated by the image d log ( K q M / p n ) d log ⁡ K q M / p n d log(K_(q)^(M)//p^(n))d \log \left(\mathcal{K}_{q}^{M} / p^{n}\right)dlog⁡(KqM/pn).
Theorem 6.2 ([28, COROLLARY 2.8]). Let F F FFF be a field of characteristic p > 0 p > 0 p > 0p>0p>0. Then the map d log : K q M ( F ) / p n W n Ω q ( F ) d log : K q M ( F ) / p n → W n Ω q ( F ) d log:K_(q)^(M)(F)//p^(n)rarrW_(n)Omega^(q)(F)d \log : K_{q}^{M}(F) / p^{n} \rightarrow W_{n} \Omega^{q}(F)dlog:KqM(F)/pn→WnΩq(F) defines an isomorphism of K q M ( F ) / p n K q M ( F ) / p n K_(q)^(M)(F)//p^(n)K_{q}^{M}(F) / p^{n}KqM(F)/pn with W n Ω log q ( F ) W n Ω log q ( F ) W_(n)Omega_(log)^(q)(F)W_{n} \Omega_{\log }^{q}(F)WnΩlogq(F).
Note that the composition
Z / p n ( q ) τ q Z / p n ( q ) K q M / p n [ q ] d log W n Ω log q [ q ] Z / p n ( q ) → Ï„ ≥ q Z / p n ( q ) ≅ K q M / p n [ − q ] → d log W n Ω log q [ − q ] Z//p^(n)(q)rarrtau_( >= q)Z//p^(n)(q)~=K_(q)^(M)//p^(n)[-q]rarr"d log"W_(n)Omega_(log)^(q)[-q]\mathbb{Z} / p^{n}(q) \rightarrow \tau_{\geq q} \mathbb{Z} / p^{n}(q) \cong \mathcal{K}_{q}^{M} / p^{n}[-q] \xrightarrow{d \log } W_{n} \Omega_{\log }^{q}[-q]Z/pn(q)→τ≥qZ/pn(q)≅KqM/pn[−q]→dlogWnΩlogq[−q]
is the map that defines the isomorphism (6.1).
The second result is a special case of the Bloch-Kato conjecture.
Theorem 6.3 (Bloch-Kato [28, THEOREM 5.12]). Let F be a henselian discretely valued field of characteristic 0 , with residue field of characteristic p > 0 p > 0 p > 0p>0p>0. Then the Galois symbol
K q M ( F ) / p n H e t q ( F , μ p n q ) K q M ( F ) / p n → H e t q F , μ p n ⊗ q K_(q)^(M)(F)//p^(n)rarrH_(et)^(q)(F,mu_(p^(n))^(ox q))K_{q}^{M}(F) / p^{n} \rightarrow H_{e t}^{q}\left(F, \mu_{p^{n}}^{\otimes q}\right)KqM(F)/pn→Hetq(F,μpn⊗q)
is an isomorphism for all n 1 . 2 n ≥ 1 . 2 n >= 1.^(2)n \geq 1 .^{2}n≥1.2
Bloch and Kato use K q M K q M K_(q)^(M)\mathcal{K}_{q}^{M}KqM to relate i R q j μ p n q i ∗ R q j ∗ μ p n ⊗ q i^(**)R^(q)j_(**)mu_(p^(n))^(ox q)i^{*} R^{q} j_{*} \mu_{p^{n}}^{\otimes q}i∗Rqj∗μpn⊗q to Ω / K q / p n Ω − / K q / p n Omega_(-//K)^(q)//p^(n)\Omega_{-/ K}^{q} / p^{n}Ω−/Kq/pn and W n Ω log q 1 W n Ω log q − 1 W_(n)Omega_(log)^(q-1)W_{n} \Omega_{\log }^{q-1}WnΩlogq−1 via the respective d log d log d logd \logdlog maps. Relying on the isomorphisms of Theorem 6.2 and Theorem 6.3, these maps from Milnor K K KKK-theory tie de Rham cohomology, crystalline cohomology and étale cohomology together, and eventually lead to a proof of Theorem 6.1.
As part of the proof, they define a surjective map
(6.2) γ : i R q j μ p n r W n Ω log q 1 (6.2) γ : i ∗ R q j ∗ μ p n ⊗ r → W n Ω log q − 1 {:(6.2)gamma:i^(**)R^(q)j_(**)mu_(p^(n))^(ox r)rarrW_(n)Omega_(log)^(q-1):}\begin{equation*} \gamma: i^{*} R^{q} j_{*} \mu_{p^{n}}^{\otimes r} \rightarrow W_{n} \Omega_{\log }^{q-1} \tag{6.2} \end{equation*}(6.2)γ:i∗Rqj∗μpn⊗r→WnΩlogq−1
on Y ét Y ét  Y_("ét ")Y_{\text {ét }}Yét  with the following property: Let θ : i j K q , ét M i R q j μ p n r θ : i ∗ j ∗ K q ,  ét  M → i ∗ R q j ∗ μ p n ⊗ r theta:i^(**)j_(**)K_(q," ét ")^(M)rarri^(**)R^(q)j_(**)mu_(p^(n))^(ox r)\theta: i^{*} j_{*} \mathcal{K}_{q, \text { ét }}^{M} \rightarrow i^{*} R^{q} j_{*} \mu_{p^{n}}^{\otimes r}θ:i∗j∗Kq, ét M→i∗Rqj∗μpn⊗r be the Galois symbol map, let u 2 , , u q u 2 , … , u q u_(2),dots,u_(q)u_{2}, \ldots, u_{q}u2,…,uq be units on X X XXX near some point y y yyy of Y Y YYY with restrictions u ¯ 1 , , u ¯ q u ¯ 1 , … , u ¯ q bar(u)_(1),dots, bar(u)_(q)\bar{u}_{1}, \ldots, \bar{u}_{q}u¯1,…,u¯q to Y Y YYY and let π Ï€ pi\piÏ€ be a parameter in Λ Î› Lambda\LambdaΛ. Then
γ θ ( { u 1 , , u q 1 , π } ) = d log ( { u ¯ 1 , , u ¯ q 1 } ) γ ∘ θ u 1 , … , u q − 1 , Ï€ = d log ⁡ u ¯ 1 , … , u ¯ q − 1 gamma@theta({u_(1),dots,u_(q-1),pi})=d log({ bar(u)_(1),dots, bar(u)_(q-1)})\gamma \circ \theta\left(\left\{u_{1}, \ldots, u_{q-1}, \pi\right\}\right)=d \log \left(\left\{\bar{u}_{1}, \ldots, \bar{u}_{q-1}\right\}\right)γ∘θ({u1,…,uq−1,Ï€})=dlog⁡({u¯1,…,u¯q−1})
We highlight this because it will be used later on in a gluing construction that defines an object of central interest for this section.
The next paper I want to mention is by Fontaine-Messing [40]. They construct a comparison isomorphism between de Rham cohomology and étale cohomology for a smooth, proper scheme X X XXX over the ring of integers O K O K O_(K)\mathcal{O}_{K}OK for K K KKK a characteristic zero local field (under some additional assumptions). The de Rham cohomology H d R q ( V / K ) H d R q ( V / K ) H_(dR)^(q)(V//K)H_{\mathrm{dR}}^{q}(V / K)HdRq(V/K) has its Hodge filtration and via the comparison isomorphism H d R q ( V / K ) H c r y s q ( Y / W ( k ) ) W ( k ) K H d R q ( V / K ) ≅ H c r y s q ( Y / W ( k ) ) ⊗ W ( k ) K H_(dR)^(q)(V//K)~=H_(crys)^(q)(Y//W(k))ox_(W(k))KH_{\mathrm{dR}}^{q}(V / K) \cong H_{\mathrm{crys}}^{q}(Y / W(k)) \otimes_{W(k)} KHdRq(V/K)≅Hcrysq(Y/W(k))⊗W(k)K H d R q ( V / K ) H d R q ( V / K ) H_(dR)^(q)(V//K)H_{\mathrm{dR}}^{q}(V / K)HdRq(V/K) acquires a Frobenius operator ϕ Ï• phi\phiÏ•; call this object H crys q ( X ) H crys  q ( X ) H_("crys ")^(q)(X)H_{\text {crys }}^{q}(X)Hcrys q(X). Fontaine-Messing construct the p p ppp-adic period ring B crys K B crys  ⊃ K B_("crys ")sup KB_{\text {crys }} \supset KBcrys ⊃K with a Galois action, a Frobenius and a filtration, and show there are isomorphisms
Fil 0 ( B crys K H crys q ( X ) ) ϕ = I d H e t q ( V K ¯ , Q p ) Fil 0 ⁡ B crys  ⊗ K H crys  q ( X ) Ï• = I d ≅ H e t q V K ¯ , Q p Fil^(0)(B_("crys ")ox_(K)H_("crys ")^(q)(X))^(phi=Id)~=H_(et)^(q)(V_( bar(K)),Q_(p))\operatorname{Fil}^{0}\left(B_{\text {crys }} \otimes_{K} H_{\text {crys }}^{q}(X)\right)^{\phi=\mathrm{Id}} \cong H_{\mathrm{et}}^{q}\left(V_{\bar{K}}, \mathbb{Q}_{p}\right)Fil0⁡(Bcrys ⊗KHcrys q(X))Ï•=Id≅Hetq(VK¯,Qp)
and
( B crys Q p H e t q ( V K ¯ , Q p ) ) G H crys q ( X ) B crys  ⊗ Q p H e t q V K ¯ , Q p G ≅ H crys  q ( X ) (B_("crys ")ox_(Q_(p))H_(et)^(q)(V_( bar(K)),Q_(p)))^(G)~=H_("crys ")^(q)(X)\left(B_{\text {crys }} \otimes_{\mathbb{Q}_{p}} H_{\mathrm{et}}^{q}\left(V_{\bar{K}}, \mathbb{Q}_{p}\right)\right)^{G} \cong H_{\text {crys }}^{q}(X)(Bcrys ⊗QpHetq(VK¯,Qp))G≅Hcrys q(X)
both compatible with the "remaining" structures.
To set this up, they consider the syntomic topology on S c h W n ( k ) S c h W n ( k ) Sch_(W_(n)(k))\mathrm{Sch}_{W_{n}(k)}SchWn(k), where a cover is a surjective syntomic map (we described syntomic maps in Section 3.3). The crystalline structure sheaf O n crys O n crys  O_(n)^("crys ")\mathcal{O}_{n}^{\text {crys }}Oncrys  defines a sheaf for the syntomic topology with a surjection to the usual structure sheaf O n O n O_(n)\mathcal{O}_{n}On on Sch W n ( k ) Sch W n ( k ) Sch_(W_(n)(k))\operatorname{Sch}_{W_{n}(k)}SchWn(k). Letting J n J n J_(n)J_{n}Jn denote the kernel of O n crys O n O n crys  → O n O_(n)^("crys ")rarrO_(n)\mathcal{O}_{n}^{\text {crys }} \rightarrow \mathcal{O}_{n}Oncrys →On, one has the r r rrr th divided power J n [ r ] J n [ r ] J_(n)^([r])J_{n}^{[r]}Jn[r]; this gives us the sheaf S ~ n r S ~ n r tilde(S)_(n)^(r)\tilde{S}_{n}^{r}S~nr defined as the kernel of ϕ p r : J n [ r ] O n crys Ï• − p r : J n [ r ] → O n crys  phi-p^(r):J_(n)^([r])rarrO_(n)^("crys ")\phi-p^{r}: J_{n}^{[r]} \rightarrow \mathcal{O}_{n}^{\text {crys }}ϕ−pr:Jn[r]→Oncrys . Modifying this by taking the image S n r S n r S_(n)^(r)S_{n}^{r}Snr of the reduction map S ~ n + r r S ~ n r S ~ n + r r → S ~ n r tilde(S)_(n+r)^(r)rarr tilde(S)_(n)^(r)\tilde{S}_{n+r}^{r} \rightarrow \tilde{S}_{n}^{r}S~n+rr→S~nr gives the inverse system { S n r } n S n r n {S_(n)^(r)}_(n)\left\{S_{n}^{r}\right\}_{n}{Snr}n and the cohomology
H ( X ¯ , S Q p r ) := ( lim n H ( X ¯ , S n r ) ) Z p Q p H ∗ X ¯ , S Q p r := lim n ←   H ∗ X ¯ , S n r ⊗ Z p Q p H^(**)(( bar(X)),S_(Q_(p))^(r)):=(lim_(n^( larr))H^(**)(( bar(X)),S_(n)^(r)))ox_(Z_(p))Q_(p)H^{*}\left(\bar{X}, S_{\mathbb{Q}_{p}}^{r}\right):=\left(\lim _{\overleftarrow{n}} H^{*}\left(\bar{X}, S_{n}^{r}\right)\right) \otimes_{\mathbb{Z}_{p}} \mathbb{Q}_{p}H∗(X¯,SQpr):=(limn←H∗(X¯,Snr))⊗ZpQp
The ring B crys + B crys  + B_("crys ")^(+)B_{\text {crys }}^{+}Bcrys +is defined as follows. The characteristic p p ppp ring O K ¯ / p O K ¯ / p O_( bar(K))//p\mathcal{O}_{\bar{K}} / pOK¯/p forms an inverse system via the Frobenius endomorphism; let
O b = lim Frob O K ¯ / p O b = lim  Frob  O K ¯ / p O^(b)=lim_(" Frob ")O_( bar(K))//p\mathcal{O}^{b}=\underset{\text { Frob }}{\operatorname{\operatorname {lim}}} \mathcal{O}_{\bar{K}} / pOb=lim Frob OK¯/p
a perfect characteristic p p ppp ring. We have the ring of truncated Witt vectors W n ( O b ) W n O b W_(n)(O^(b))W_{n}\left(\mathcal{O}^{\mathrm{b}}\right)Wn(Ob) and a surjection π n : W n ( O b ) O K ¯ / p n Ï€ n : W n O b → O K ¯ / p n pi_(n):W_(n)(O^(b))rarrO_( bar(K))//p^(n)\pi_{n}: W_{n}\left(\mathcal{O}^{b}\right) \rightarrow \mathcal{O}_{\bar{K}} / p^{n}Ï€n:Wn(Ob)→OK¯/pn. Let W n D P ( O K ¯ ) W n D P O K ¯ W_(n)^(DP)(O_( bar(K)))W_{n}^{\mathrm{DP}}\left(\mathcal{O}_{\bar{K}}\right)WnDP(OK¯) be the divided power envelope of the kernel of π n Ï€ n pi_(n)\pi_{n}Ï€n, forming the inverse system { W n D P ( O K ¯ ) } n 0 W n D P O K ¯ n ≥ 0 {W_(n)^(DP)(O_( bar(K)))}_(n >= 0)\left\{W_{n}^{\mathrm{DP}}\left(\mathcal{O}_{\bar{K}}\right)\right\}_{n \geq 0}{WnDP(OK¯)}n≥0. Let
B crys + := K W ( k ) lim n W n D P ( O K ¯ ) B crys  + := K ⊗ W ( k ) lim n ←   W n D P O K ¯ B_("crys ")^(+):=Kox_(W(k))lim_(n^( larr))W_(n)^(DP)(O_( bar(K)))B_{\text {crys }}^{+}:=K \otimes_{W(k)} \lim _{\overleftarrow{n}} W_{n}^{\mathrm{DP}}\left(\mathcal{O}_{\bar{K}}\right)Bcrys +:=K⊗W(k)limn←WnDP(OK¯)
The Frobenius on W n ( O b ) W n O b W_(n)(O^(b))W_{n}\left(\mathcal{O}^{\mathrm{b}}\right)Wn(Ob) induces a Frobenius on B crys + B crys  + B_("crys ")^(+)B_{\text {crys }}^{+}Bcrys +and the filtration J n [ ] J n [ ∗ ] J_(n)^([**])J_{n}^{[*]}Jn[∗] of W n D P ( O K ¯ ) W n D P O K ¯ W_(n)^(DP)(O_( bar(K)))W_{n}^{\mathrm{DP}}\left(\mathcal{O}_{\bar{K}}\right)WnDP(OK¯) induces a filtration Fil* B crys + B crys  + B_("crys ")^(+)B_{\text {crys }}^{+}Bcrys +on B crys + B crys  + B_("crys ")^(+)B_{\text {crys }}^{+}Bcrys +.
The derived push-forward of the complex J n [ r ] ϕ p r O n crys J n [ r ] → Ï• − p r O n crys  J_(n)^([r])rarr"phi-p^(r)"O_(n)^("crys ")J_{n}^{[r]} \xrightarrow{\phi-p^{r}} \mathcal{O}_{n}^{\text {crys }}Jn[r]→ϕ−prOncrys  is an analog of the Deligne complex, as expressed in the following theorem.
Theorem 6.4 ([40, corollary to theorem 1.6, LemMA 3.1]). Suppose that X X XXX is admissible 3 3 ^(3){ }^{3}3 and Λ = W ( k ) Λ = W ( k ) Lambda=W(k)\Lambda=W(k)Λ=W(k). Then for m r < p m ≤ r < p m <= r < pm \leq r<pm≤r<p there is an exact sequence
0 H m ( X ¯ , S Q p r ) Fil r ( B crys + K H d R m ( V / K ) ) ϕ p r H d R ( V / K ) 0 0 → H m X ¯ , S Q p r → Fil r ⁡ B crys  + ⊗ K H d R m ( V / K ) → Ï• − p r H d R ( V / K ) → 0 0rarrH^(m)(( bar(X)),S_(Q_(p))^(r))rarrFil^(r)(B_("crys ")^(+)ox_(K)H_(dR)^(m)(V//K))rarr"phi-p^(r)"H_(dR)(V//K)rarr00 \rightarrow H^{m}\left(\bar{X}, S_{\mathbb{Q}_{p}}^{r}\right) \rightarrow \operatorname{Fil}^{r}\left(B_{\text {crys }}^{+} \otimes_{K} H_{\mathrm{dR}}^{m}(V / K)\right) \xrightarrow{\phi-p^{r}} H_{\mathrm{dR}}(V / K) \rightarrow 00→Hm(X¯,SQpr)→Filr⁡(Bcrys +⊗KHdRm(V/K))→ϕ−prHdR(V/K)→0
In other words,
H m ( X ¯ , S Q p r ) = ( Fil r ( B crys + K H d R m ( V / K ) ) ) ϕ = p r H m X ¯ , S Q p r =  Fil  r B crys  + ⊗ K H d R m ( V / K ) Ï• = p r H^(m)(( bar(X)),S_(Q_(p))^(r))=(" Fil "^(r)(B_("crys ")^(+)ox_(K)H_(dR)^(m)(V//K)))^(phi=p^(r))H^{m}\left(\bar{X}, S_{\mathbb{Q}_{p}}^{r}\right)=\left(\text { Fil }^{r}\left(B_{\text {crys }}^{+} \otimes_{K} H_{\mathrm{dR}}^{m}(V / K)\right)\right)^{\phi=p^{r}}Hm(X¯,SQpr)=( Fil r(Bcrys +⊗KHdRm(V/K)))Ï•=pr
3 See [ 40 , $ 2 . 1 ] [ 40 , $ 2 . 1 ] [40,$2.1][40, \mathbf{\$ 2 . 1}][40,$2.1] for the definition of admissible X X XXX.
To involve étale cohomology in the picture, Fontaine-Messing introduce the syntomic-étale site on formal Spf ( W ( k ) ) Spf ⁡ ( W ( k ) ) Spf(W(k))\operatorname{Spf}(W(k))Spf⁡(W(k))-schemes, where a cover is a map { U n } n { V n } n U n n → V n n {U_(n)}_(n)rarr{V_(n)}_(n)\left\{U_{n}\right\}_{n} \rightarrow\left\{V_{n}\right\}_{n}{Un}n→{Vn}n such that U n V n U n → V n U_(n)rarrV_(n)U_{n} \rightarrow V_{n}Un→Vn is a syntomic cover for all n n nnn and is an étale cover on the rigid analytic generic fibers. This extends to the syntomic-étale site on X ¯ X ¯ bar(X)\bar{X}X¯, where an object is U X ¯ U → X ¯ U rarr bar(X)U \rightarrow \bar{X}U→X¯, quasi-finite and syntomic, with U K ¯ V K ¯ U K ¯ → V K ¯ U_( bar(K))rarrV_( bar(K))U_{\bar{K}} \rightarrow V_{\bar{K}}UK¯→VK¯ étale. Letting Z Z Z\mathcal{Z}Z be the formal completion of X ¯ X ¯ bar(X)\bar{X}X¯, we have the diagram of sites
Z syn-ét i X ¯ syn-ét j V K ¯ , ét Z syn-ét  → i X ¯ syn-ét  ← j V K ¯ ,  ét  Z_("syn-ét ")rarr^(i) bar(X)_("syn-ét ")larr^(j)V_( bar(K)," ét ")Z_{\text {syn-ét }} \stackrel{i}{\rightarrow} \bar{X}_{\text {syn-ét }} \stackrel{j}{\leftarrow} V_{\bar{K}, \text { ét }}Zsyn-ét →iX¯syn-ét ←jVK¯, ét 
Fontaine-Messing prove a patching result, that a sheaf on X ¯ syn-ét X ¯ syn-ét  bar(X)_("syn-ét ")\bar{X}_{\text {syn-ét }}X¯syn-ét  is given by a triple ( F , E , α ) ( F , E , α ) (F,E,alpha)(\mathcal{F}, \mathscr{E}, \alpha)(F,E,α), with F F F\mathscr{F}F a sheaf on Z syn-ét , E Z syn-ét  , E Z_("syn-ét "),E\mathcal{Z}_{\text {syn-ét }}, \mathcal{E}Zsyn-ét ,E a sheaf on V K ¯ , e t V K ¯ , e t V_( bar(K),et)V_{\bar{K}, \mathrm{et}}VK¯,et, and α : F i R j E α : F → i ∗ R j ∗ E alpha:Frarri^(**)Rj_(**)E\alpha: \mathscr{F} \rightarrow i^{*} R j_{*} \mathscr{E}α:F→i∗Rj∗E a morphism. Using this description, they construct a sheaf on X ¯ syn-ét X ¯ syn-ét  bar(X)_("syn-ét ")\bar{X}_{\text {syn-ét }}X¯syn-ét  by defining a certain morphism (see [ 40 , § 5.1 ] ) [ 40 , § 5.1 ] ) [40,§5.1])[40, \S 5.1])[40,§5.1])
α : S n r i R j μ p n r α : S n r → i ∗ R j ∗ μ p n ⊗ r alpha:S_(n)^(r)rarri^(**)Rj_(**)mu_(p^(n))^(ox r)\alpha: S_{n}^{r} \rightarrow i^{*} R j_{*} \mu_{p^{n}}^{\otimes r}α:Snr→i∗Rj∗μpn⊗r
The resulting sheaf S ~ n r S ~ n r tilde(S)_(n)^(r)\tilde{S}_{n}^{r}S~nr has
j S ~ n r μ p n r , i S ~ n r S n r j ∗ S ~ n r ≅ μ p n ⊗ r , i ∗ S ~ n r ≅ S n r j^(**) tilde(S)_(n)^(r)~=mu_(p^(n))^(ox r),quadi^(**) tilde(S)_(n)^(r)~=S_(n)^(r)j^{*} \tilde{S}_{n}^{r} \cong \mu_{p^{n}}^{\otimes r}, \quad i^{*} \tilde{S}_{n}^{r} \cong S_{n}^{r}j∗S~nr≅μpn⊗r,i∗S~nr≅Snr
It follows from proper base-change (see the proof of [40, PROPOSITION 6.2]) that the restriction map H ( X ¯ syn-é , S ~ n r ) H ( Z syn-et , S n r ) H ∗ X ¯ syn-é  , S ~ n r → H ∗ Z syn-et  , S n r H^(**)( bar(X)_("syn-é "), tilde(S)_(n)^(r))rarrH^(**)(Z_("syn-et "),S_(n)^(r))H^{*}\left(\bar{X}_{\text {syn-é }}, \tilde{S}_{n}^{r}\right) \rightarrow H^{*}\left(\mathcal{Z}_{\text {syn-et }}, S_{n}^{r}\right)H∗(X¯syn-é ,S~nr)→H∗(Zsyn-et ,Snr) is an isomorphism, and we also have
( lim n H ( Z s y n e t , S n r ) ) Z p Q p ( Fil r ( B c r y s + K H d R m ( V / K ) ) ) ϕ = p r lim n ←   H ∗ Z s y n − e t , S n r ⊗ Z p Q p ≅ Fil r ⁡ B c r y s + ⊗ K H d R m ( V / K ) Ï• = p r (lim_(n^( larr))H^(**)(Z_(syn-et),S_(n)^(r)))ox_(Z_(p))Q_(p)~=(Fil^(r)(B_(crys)^(+)ox_(K)H_(dR)^(m)(V//K)))^(phi=p^(r))\left(\lim _{\overleftarrow{n}} H^{*}\left(\mathcal{Z}_{\mathrm{syn}-\mathrm{et}}, S_{n}^{r}\right)\right) \otimes_{\mathbb{Z}_{p}} \mathbb{Q}_{p} \cong\left(\operatorname{Fil}^{r}\left(B_{\mathrm{crys}}^{+} \otimes_{K} H_{\mathrm{dR}}^{m}(V / K)\right)\right)^{\phi=p^{r}}(limn←H∗(Zsyn−et,Snr))⊗ZpQp≅(Filr⁡(Bcrys+⊗KHdRm(V/K)))Ï•=pr
Moreover, the restriction map j j ∗ j^(**)j^{*}j∗ gives
j : H ( X ¯ s y n e t , S ~ n r ) H e t ( V K ¯ , μ p n r ) j ∗ : H ∗ X ¯ s y n − e t , S ~ n r → H e t ∗ V K ¯ , μ p n ⊗ r j^(**):H^(**)( bar(X)_(syn-et), tilde(S)_(n)^(r))rarrH_(et)^(**)(V_( bar(K)),mu_(p^(n))^(ox r))j^{*}: H^{*}\left(\bar{X}_{\mathrm{syn}-\mathrm{et}}, \tilde{S}_{n}^{r}\right) \rightarrow H_{\mathrm{et}}^{*}\left(V_{\bar{K}}, \mu_{p^{n}}^{\otimes r}\right)j∗:H∗(X¯syn−et,S~nr)→Het∗(VK¯,μpn⊗r)
so passing to the limit, we have the map
β : ( Fil r ( B crys + K H d R m ( V / K ) ) ) ϕ = p r H e t m ( V K ¯ , Q p ( r ) ) β : Fil r ⁡ B crys  + ⊗ K H d R m ( V / K ) Ï• = p r → H e t m V K ¯ , Q p ( r ) beta:(Fil^(r)(B_("crys ")^(+)ox_(K)H_(dR)^(m)(V//K)))^(phi=p^(r))rarrH_(et)^(m)(V_( bar(K)),Q_(p)(r))\beta:\left(\operatorname{Fil}^{r}\left(B_{\text {crys }}^{+} \otimes_{K} H_{\mathrm{dR}}^{m}(V / K)\right)\right)^{\phi=p^{r}} \rightarrow H_{\mathrm{et}}^{m}\left(V_{\bar{K}}, \mathbb{Q}_{p}(r)\right)β:(Filr⁡(Bcrys +⊗KHdRm(V/K)))Ï•=pr→Hetm(VK¯,Qp(r))
which they show is an isomorphism.
This gives a twisted version of the result announced at the beginning of our discussion. To recover the untwisted version, they define a map
Q p ( 1 ) ( K ¯ ) B crys + Q p ( 1 ) ( K ¯ ) → B crys  + Q_(p)(1)( bar(K))rarrB_("crys ")^(+)\mathbb{Q}_{p}(1)(\bar{K}) \rightarrow B_{\text {crys }}^{+}Qp(1)(K¯)→Bcrys +
by sending a p n p n p^(n)p^{n}pn-root of unity ε ε epsi\varepsilonε in K ¯ K ¯ bar(K)\bar{K}K¯ to the logarithm of the Teichmüller lift of the mod p p ppp reduction of ε ε epsi\varepsilonε, and passing to the limit in n n nnn. Let t Q p ( 1 ) ( K ¯ ) t ∈ Q p ( 1 ) ( K ¯ ) t inQ_(p)(1)( bar(K))t \in \mathbb{Q}_{p}(1)(\bar{K})t∈Qp(1)(K¯) be a nonzero element and define B crys = B crys + [ 1 / t ] B crys  = B crys  + [ 1 / t ] B_("crys ")=B_("crys ")^(+)[1//t]B_{\text {crys }}=B_{\text {crys }}^{+}[1 / t]Bcrys =Bcrys +[1/t], with induced filtration and Galois action. Twisting with respect to t t ttt translates the twisted version to the untwisted one.
The sheaf S ~ n r S ~ n r tilde(S)_(n)^(r)\tilde{S}_{n}^{r}S~nr is only defined on X ¯ syn-ét X ¯ syn-ét  bar(X)_("syn-ét ")\bar{X}_{\text {syn-ét }}X¯syn-ét  for X X XXX smooth over S S SSS and for r < p r < p r < pr<pr<p, and with base-ring Λ Î› Lambda\LambdaΛ equal to W ( k ) W ( k ) W(k)W(k)W(k), i.e., in the unramified case. Kato [79] studies the derived push-forward S n ( r ) S n ( r ) S_(n)(r)S_{n}(r)Sn(r) of the syntomic sheaf S ~ n r S ~ n r tilde(S)_(n)^(r)\tilde{S}_{n}^{r}S~nr to S m k , et S m k ,  et  Sm_(k," et ")\mathrm{Sm}_{k, \text { et }}Smk, et . Kurihara [82, § 1 § 1 §1\S 1§1, THEOREM] considers the ramified case and also clarifies the relation of S n ( r ) S n ( r ) S_(n)(r)S_{n}(r)Sn(r) with the sheaf of log log log\loglog forms W n Ω log r 1 W n Ω log r − 1 W_(n)Omega_(log)^(r-1)W_{n} \Omega_{\log }^{r-1}WnΩlogr−1
Theorem 6.5 (Kurihara). Suppose that [ k : k p ] < k : k p < ∞ [k:k^(p)] < oo\left[k: k^{p}\right]<\infty[k:kp]<∞. Let X S X → S X rarr SX \rightarrow SX→S be smooth and projective and suppose that r < p 1 r < p − 1 r < p-1r<p-1r<p−1. Then there is a distinguished triangle in D ( Y e t ^ ) D Y e t ^ D(Y_( hat(et)))D\left(Y_{\hat{e t}}\right)D(Yet^),
W n Ω log r 1 [ r 1 ] S n ( r ) i R j μ p n r W n Ω log r 1 [ r ] W n Ω log r − 1 [ − r − 1 ] → S n ( r ) → i ∗ R j ∗ μ p n ⊗ r → W n Ω log r − 1 [ − r ] W_(n)Omega_(log)^(r-1)[-r-1]rarrS_(n)(r)rarri^(**)Rj_(**)mu_(p^(n))^(ox r)rarrW_(n)Omega_(log)^(r-1)[-r]W_{n} \Omega_{\log }^{r-1}[-r-1] \rightarrow S_{n}(r) \rightarrow i^{*} R j_{*} \mu_{p^{n}}^{\otimes r} \rightarrow W_{n} \Omega_{\log }^{r-1}[-r]WnΩlogr−1[−r−1]→Sn(r)→i∗Rj∗μpn⊗r→WnΩlogr−1[−r]
Schneider [108] extends the construction of S n ( r ) S n ( r ) S_(n)(r)S_{n}(r)Sn(r) to all r 0 r ≥ 0 r >= 0r \geq 0r≥0 by using the BlochKato symbol map γ γ gamma\gammaγ of (6.2) to give a map s : τ r R j μ p n r i W n Ω log r 1 s : Ï„ ≤ r R j ∗ μ p n ⊗ r → i ∗ W n Ω log r − 1 s:tau_( <= r)Rj_(**)mu_(p^(n))^(ox r)rarri_(**)W_(n)Omega_(log)^(r-1)s: \tau_{\leq r} R j_{*} \mu_{p^{n}}^{\otimes r} \rightarrow i_{*} W_{n} \Omega_{\log }^{r-1}s:τ≤rRj∗μpn⊗r→i∗WnΩlogr−1 with i s i ∗ s i^(**)si^{*} si∗s the composition
i τ r R j μ p n r i R r j μ p n r [ r ] γ W n Ω log r 1 [ r ] . i ∗ Ï„ ≤ r R j ∗ μ p n ⊗ r → i ∗ R r j ∗ μ p n ⊗ r [ − r ] → γ W n Ω log r − 1 [ − r ] . i^(**)tau_( <= r)Rj_(**)mu_(p^(n))^(ox r)rarri^(**)R^(r)j_(**)mu_(p^(n))^(ox r)[-r]rarr"gamma"W_(n)Omega_(log)^(r-1)[-r].i^{*} \tau_{\leq r} R j_{*} \mu_{p^{n}}^{\otimes r} \rightarrow i^{*} R^{r} j_{*} \mu_{p^{n}}^{\otimes r}[-r] \xrightarrow{\gamma} W_{n} \Omega_{\log }^{r-1}[-r] .i∗τ≤rRj∗μpn⊗r→i∗Rrj∗μpn⊗r[−r]→γWnΩlogr−1[−r].
Schneider then defines the sheaf S n ( r ) S n ( r ) S_(n)(r)S_{n}(r)Sn(r) as Cone ( s ) [ 1 ] ( s ) [ − 1 ] (s)[-1](s)[-1](s)[−1], giving the distinguished triangle
(6.3) i W n Ω log r 1 [ r 1 ] S n ( r ) R j μ p n r s i W n Ω log r 1 [ r ] , (6.3) i ∗ W n Ω log r − 1 [ − r − 1 ] → S n ( r ) → R j ∗ μ p n ⊗ r → s i ∗ W n Ω log r − 1 [ − r ] , {:(6.3)i_(**)W_(n)Omega_(log)^(r-1)[-r-1]rarrS_(n)(r)rarr Rj_(**)mu_(p^(n))^(ox r)rarr"s"i_(**)W_(n)Omega_(log)^(r-1)[-r]",":}\begin{equation*} i_{*} W_{n} \Omega_{\log }^{r-1}[-r-1] \rightarrow S_{n}(r) \rightarrow R j_{*} \mu_{p^{n}}^{\otimes r} \xrightarrow{s} i_{*} W_{n} \Omega_{\log }^{r-1}[-r], \tag{6.3} \end{equation*}(6.3)i∗WnΩlogr−1[−r−1]→Sn(r)→Rj∗μpn⊗r→si∗WnΩlogr−1[−r],
which recovers the one in Kurihara's theorem for r < p 1 r < p − 1 r < p-1r<p-1r<p−1 by applying i i ∗ i^(**)i^{*}i∗.
Using a similar method, Schneider's construction was extended to the semi-stable case by Sato [106], who defines the object T n ( r ) D ( X et ) T n ( r ) ∈ D X et  T_(n)(r)in D(X_("et "))\mathfrak{T}_{n}(r) \in D\left(X_{\text {et }}\right)Tn(r)∈D(Xet ) with T n ( r ) S n ( r ) T n ( r ) ≅ S n ( r ) T_(n)(r)~=S_(n)(r)\mathfrak{T}_{n}(r) \cong S_{n}(r)Tn(r)≅Sn(r) in the smooth case.

6.2. Étale motivic cohomology

We return to algebraic cycles. As before, we consider a smooth separated finite type S S SSS-scheme X S = Spec Λ X → S = Spec ⁡ Λ X rarr S=Spec LambdaX \rightarrow S=\operatorname{Spec} \LambdaX→S=Spec⁡Λ with generic fiber j : V X j : V → X j:V rarr Xj: V \rightarrow Xj:V→X and special fiber i : Y X i : Y → X i:Y rarr Xi: Y \rightarrow Xi:Y→X, and with Λ Î› Lambda\LambdaΛ a mixed characteristic ( 0 , p ) ( 0 , p ) (0,p)(0, p)(0,p) dvr with perfect residue field.
Geisser [49] considers the motivic complex Z ( r ) X Z ( r ) X Z(r)_(X)\mathbb{Z}(r)_{X}Z(r)X on a smooth S S SSS-scheme X S X → S X rarr SX \rightarrow SX→S as a sheaf of complexes on X N i s X N i s X_(Nis)X_{\mathrm{Nis}}XNis. Here we use the reindexed Bloch cycle complex to define Z ( r ) X ( U ) Z ( r ) X ∗ ( U ) Z(r)_(X)^(**)(U)\mathbb{Z}(r)_{X}^{*}(U)Z(r)X∗(U) as
Z ( r ) X ( U ) := z r ( U , 2 r ) Z ( r ) X ∗ ( U ) := z r ( U , 2 r − ∗ ) Z(r)_(X)^(**)(U):=z^(r)(U,2r-**)\mathbb{Z}(r)_{X}^{*}(U):=z^{r}(U, 2 r-*)Z(r)X∗(U):=zr(U,2r−∗)
and define the motivic complexes Z ( r ) V Z ( r ) V Z(r)_(V)\mathbb{Z}(r)_{V}Z(r)V and Z ( r ) Y Z ( r ) Y Z(r)_(Y)\mathbb{Z}(r)_{Y}Z(r)Y on V V VVV and Y Y YYY similarly.
Let α : ( ) êt ( ) Nis α : ( − ) êt  → ( − ) Nis  alpha:(-)_("êt ")rarr(-)_("Nis ")\alpha:(-)_{\text {êt }} \rightarrow(-)_{\text {Nis }}α:(−)êt →(−)Nis  be the change of topology map. Sheafifying for the étale topology gives complexes Z ( r ) e t , X , Z ( r ) e t , V Z ( r ) e t , X , Z ( r ) e t , V Z(r)_(et,X),Z(r)_(et,V)\mathbb{Z}(r)_{\mathrm{et}, X}, \mathbb{Z}(r)_{\mathrm{et}, V}Z(r)et,X,Z(r)et,V, and Z ( r ) e t , Y Z ( r ) e t , Y Z(r)_(et,Y)\mathbb{Z}(r)_{\mathrm{et}, Y}Z(r)et,Y. Geisser shows that various known properties of Z ( r ) X , Z ( r ) V Z ( r ) X , Z ( r ) V Z(r)_(X),Z(r)_(V)\mathbb{Z}(r)_{X}, \mathbb{Z}(r)_{V}Z(r)X,Z(r)V, and Z ( r ) Y Z ( r ) Y Z(r)_(Y)\mathbb{Z}(r)_{Y}Z(r)Y, such as the purity isomorphism [84, THEOREM 1.7]
i ! Z ( r ) X Z ( r 1 ) Y [ 2 ] i ! Z ( r ) X ≅ Z ( r − 1 ) Y [ − 2 ] i^(!)Z(r)_(X)~=Z(r-1)_(Y)[-2]i^{!} \mathbb{Z}(r)_{X} \cong \mathbb{Z}(r-1)_{Y}[-2]i!Z(r)X≅Z(r−1)Y[−2]
the theorem of Geisser-Levine [52]
Z / p n ( r ) Y W n Ω log , Y r [ r ] Z / p n ( r ) Y ≅ W n Ω log , Y r [ − r ] Z//p^(n)(r)_(Y)~=W_(n)Omega_(log,Y)^(r)[-r]\mathbb{Z} / p^{n}(r)_{Y} \cong W_{n} \Omega_{\log , Y}^{r}[-r]Z/pn(r)Y≅WnΩlog,Yr[−r]
the Suslin-Voevodsky isomorphism in D b ( V êt ) D b V êt  D^(b)(V_("êt "))D^{b}\left(V_{\text {êt }}\right)Db(Vêt ) (Beilinson's axiom (iv)(a))
j Z / p n ( r ) e t , X Z / p n ( n ) e t , V μ p n r j ∗ Z / p n ( r ) e t , X ≅ Z / p n ( n ) e t , V ≅ μ p n ⊗ r j^(**)Z//p^(n)(r)_(et,X)~=Z//p^(n)(n)_(et,V)~=mu_(p^(n))^(ox r)j^{*} \mathbb{Z} / p^{n}(r)_{\mathrm{e} t, X} \cong \mathbb{Z} / p^{n}(n)_{\mathrm{e} t, V} \cong \mu_{p^{n}}^{\otimes r}j∗Z/pn(r)et,X≅Z/pn(n)et,V≅μpn⊗r
and the Beilinson-Lichtenbaum conjectures (now a theorem)
Z / p n ( r ) V τ r R α μ p n r , R r + 1 α Z ( r ) e t , V = 0 Z / p n ( r ) V ≅ Ï„ ≤ r R α ∗ μ p n ⊗ r , R r + 1 α ∗ Z ( r ) e t , V = 0 Z//p^(n)(r)_(V)~=tau_( <= r)Ralpha_(**)mu_(p^(n))^(ox r),quadR^(r+1)alpha_(**)Z(r)_(et,V)=0\mathbb{Z} / p^{n}(r)_{V} \cong \tau_{\leq r} R \alpha_{*} \mu_{p^{n}}^{\otimes r}, \quad R^{r+1} \alpha_{*} \mathbb{Z}(r)_{\mathfrak{e t}, V}=0Z/pn(r)V≅τ≤rRα∗μpn⊗r,Rr+1α∗Z(r)et,V=0
have as consequence
Theorem 6.6 (Geisser [49, THEOREM 1.3]). Let X X → X rarrX \rightarrowX→ Spec Λ Î› Lambda\LambdaΛ be smooth and essentially of finite type, with Λ Î› Lambda\LambdaΛ a complete discrete valuation ring of mixed characteristic ( 0 , p ) ( 0 , p ) (0,p)(0, p)(0,p). Then there is a distinguished triangle in D b ( X èt ) D b X èt  D^(b)(X_("èt "))D^{b}\left(X_{\text {èt }}\right)Db(Xèt ),
i W n Ω log r 1 [ r 1 ] Z / p n ( r ) e ́ t τ r R j μ p n r i W n Ω log r 1 [ r ] i ∗ W n Ω log r − 1 [ − r − 1 ] → Z / p n ( r ) e ́ t → Ï„ ≤ r R j ∗ μ p n ⊗ r → i ∗ W n Ω log r − 1 [ − r ] i_(**)W_(n)Omega_(log)^(r-1)[-r-1]rarrZ//p^(n)(r)_(ét)rarrtau_( <= r)Rj_(**)mu_(p^(n))^(ox r)rarri_(**)W_(n)Omega_(log)^(r-1)[-r]i_{*} W_{n} \Omega_{\log }^{r-1}[-r-1] \rightarrow \mathbb{Z} / p^{n}(r)_{e ́ t} \rightarrow \tau_{\leq r} R j_{*} \mu_{p^{n}}^{\otimes r} \rightarrow i_{*} W_{n} \Omega_{\log }^{r-1}[-r]i∗WnΩlogr−1[−r−1]→Z/pn(r)ét→τ≤rRj∗μpn⊗r→i∗WnΩlogr−1[−r]
and an isomorphism Z / p n ( r ) e ́ t S n ( r ) Z / p n ( r ) e ́ t ≅ S n ( r ) Z//p^(n)(r)_(ét)~=S_(n)(r)\mathbb{Z} / p^{n}(r)_{e ́ t} \cong S_{n}(r)Z/pn(r)ét≅Sn(r) in D b ( X e ˙ t ) D b X e Ë™ t D^(b)(X_(e^(Ë™)t))D^{b}\left(X_{\dot{e} t}\right)Db(XeË™t) that transforms this triangle to Schneider's defining triangle (6.3).
Zhong has extended this to the semi-stable case, establishing an isomorphism with Sato's construction I n ( r ) I n ( r ) I_(n)(r)\mathfrak{I}_{n}(r)In(r) after a truncation [128, PROPOSITION 4.5]:
τ r Z / p n ( r ) ét T n ( r ) Ï„ ≤ r Z / p n ( r ) ét  ≅ T n ( r ) tau_( <= r)Z//p^(n)(r)_("ét ")~=T_(n)(r)\tau_{\leq r} \mathbb{Z} / p^{n}(r)_{\text {ét }} \cong \mathfrak{T}_{n}(r)τ≤rZ/pn(r)ét ≅Tn(r)
Assuming a "weak Gersten conjecture" for Z / p n ( r ) êt Z / p n ( r ) êt  Z//p^(n)(r)_("êt ")\mathbb{Z} / p^{n}(r)_{\text {êt }}Z/pn(r)êt , the truncation is removed [128, tHEoREM 4.8].

6.3. The theorems of Geisser-Hesselholt

The construction of a motivic tower for integral p p ppp-adic Hodge theory by BhattMorrow-Scholze relies on properties of p p ppp-completed topological cyclic homology, including the results of Geisser-Hesselholt identifying this with the p p ppp-completed étale K K KKK-theory. We give a brief résumé of these constructions. Fix as before our mixed characteristic dvr Λ dvr ⁡ Λ dvr Lambda\operatorname{dvr} \Lambdadvr⁡Λ with perfect residue field k k kkk.
Topological cyclic homology for a fixed prime p p ppp is a spectrum refined version of Connes' cyclic homology and is defined for a scheme X X XXX with a topology τ { Ï„ ∈ { tau in{\tau \in\{τ∈{ ét, Nis, Zar}; we use the étale topology throughout. There is an inverse system of spectra { TC m ( X , p ) } m N TC m ⁡ ( X , p ) m ∈ N {TC^(m)(X,p)}_(m inN)\left\{\operatorname{TC}^{m}(X, p)\right\}_{m \in \mathbb{N}}{TCm⁡(X,p)}m∈N defining T C ( X ; p ) T C ( X ; p ) TC(X;p)\mathrm{TC}(X ; p)TC(X;p) as the homotopy inverse limit
T C ( X ; p ) := holim m T C m ( X , p ) T C ( X ; p ) := holim m T C m ( X , p ) TC(X;p):=holim _(m)TC^(m)(X,p)\mathrm{TC}(X ; p):=\underset{m}{\operatorname{holim}} \mathrm{TC}^{m}(X, p)TC(X;p):=holimmTCm(X,p)
Let T C i T C i TC_(i)\mathcal{T} \mathscr{C}_{i}TCi denote the étale sheaf associated to the presheaf of the i i iii th pro-homotopy groups U π i T C ( U ; p ) U ↦ Ï€ i T C â‹… ( U ; p ) U|->pi_(i)TC*(U;p)U \mapsto \pi_{i} \mathrm{TC} \cdot(U ; p)U↦πiTCâ‹…(U;p). There is a descent spectral sequence
E 2 s , t = H c o n t s ( X , T e t ) T C s t ( X ; p ) E 2 s , t = H c o n t s X , T e − t ⋅ ⇒ T C − s − t ( X ; p ) E_(2)^(s,t)=H_(cont)^(s)(X,Te_(-t)^(*))=>TC_(-s-t)(X;p)E_{2}^{s, t}=H_{\mathrm{cont}}^{s}\left(X, \mathcal{T} \mathcal{e}_{-t}^{\cdot}\right) \Rightarrow \mathrm{TC}_{-s-t}(X ; p)E2s,t=Hconts(X,Te−t⋅)⇒TC−s−t(X;p)
and a cyclotomic trace map
trc : K ( X ) T C ( X ; p ) trc : K ( X ) → T C ( X ; p ) trc:K(X)rarrTC(X;p)\operatorname{trc}: K(X) \rightarrow \mathrm{TC}(X ; p)trc:K(X)→TC(X;p)
Let k k kkk be a perfect field of characteristic p > 0 p > 0 p > 0p>0p>0. It follows from a result of Hesselholt [60, THEOREM B] that there is a isomorphism of pro-sheaves on S m k ét S m k  ét  Sm_(k" ét ")\mathrm{Sm}_{k \text { ét }}Smk ét 
(6.4) T C r W Ω log r (6.4) T C r ≅ W â‹… Ω log r {:(6.4)TC_(r)~=W*Omega_(log)^(r):}\begin{equation*} \mathcal{T} C_{r} \cong W \cdot \Omega_{\log }^{r} \tag{6.4} \end{equation*}(6.4)TCr≅W⋅Ωlogr
The map trc induces the map of pro-sheaves on S m k ét S m k  ét  Sm_(k" ét ")\mathrm{Sm}_{k \text { ét }}Smk ét 
trc : K i ( Z / p ) T C i trc : K i Z / p ⋅ → T C i trc:K_(i)(Z//p^(*))rarrTC_(i)\operatorname{trc}: \mathcal{K}_{i}\left(\mathbb{Z} / p^{\cdot}\right) \rightarrow \mathcal{T} \mathscr{C}_{i}trc:Ki(Z/p⋅)→TCi
where K i ( Z / p ) K i Z / p â‹… K_(i)(Z//p^(*))\mathcal{K}_{i}\left(\mathbb{Z} / p^{\cdot}\right)Ki(Z/pâ‹…) is the pro-étale sheaf associated to the system of presheaves U U ↦ U|->U \mapstoU↦ { K i ( U , Z / p ν ) } ν K i U , Z / p ν ν {K_(i)(U,Z//p^(nu))}_(nu)\left\{K_{i}\left(U, \mathbb{Z} / p^{\nu}\right)\right\}_{\nu}{Ki(U,Z/pν)}ν. Relying on the main theorem of [52] and the isomorphism (6.4), Geisser and Hesselholt show
Theorem 6.7 (Geisser-Hesselholt [50, COROLLARY 4.2.5, THEOREM 4.2.6]).
  1. The trace map trc : K i ( Z / p ) T Y i K i Z / p â‹… → T Y i K_(i)(Z//p^(*))rarrTY_(i)\mathcal{K}_{i}\left(\mathbb{Z} / p^{\cdot}\right) \rightarrow \mathcal{T} \mathcal{Y}_{i}Ki(Z/pâ‹…)→TYi is an isomorphism of pro-sheaves on S m k e ́ t S m k e ́ t Sm_(két)\mathrm{Sm}_{k e ́ t}Smkét.
  2. For Y Sm k , TC ( Y ; p ) Y ∈ Sm k , TC ⁡ ( Y ; p ) Y inSm_(k),TC(Y;p)Y \in \operatorname{Sm}_{k}, \operatorname{TC}(Y ; p)Y∈Smk,TC⁡(Y;p) is weakly equivalent to the p p ppp-completed étale K K KKK-theory spectrum of Y Y YYY,
K e ́ t ( Y ) p := holim m K e ́ t ( Y , Z / p n ) TC ( Y ; p ) K e ́ t ( Y ) ∧ p := holim m K e ́ t Y , Z / p n ≅ TC ⁡ ( Y ; p ) K^(ét)(Y)^(^^_(p)):=holim _(m)K^(ét)(Y,Z//p^(n))~=TC(Y;p)K^{e ́ t}(Y)^{\wedge_{p}}:=\underset{m}{\operatorname{holim}} K^{e ́ t}\left(Y, \mathbb{Z} / p^{n}\right) \cong \operatorname{TC}(Y ; p)Két(Y)∧p:=holimmKét(Y,Z/pn)≅TC⁡(Y;p)
and this weak equivalence arises from the weak equivalences at the finite level
trc : K e t ( Y , Z / p ν ) T C ( Y ; p , Z / p ν ) trc : K e t Y , Z / p ν → ∼ T C Y ; p , Z / p ν trc:K^(et)(Y,Z//p^(nu))rarr"∼"TC(Y;p,Z//p^(nu))\operatorname{trc}: K^{e t}\left(Y, \mathbb{Z} / p^{\nu}\right) \xrightarrow{\sim} \mathrm{TC}\left(Y ; p, \mathbb{Z} / p^{\nu}\right)trc:Ket(Y,Z/pν)→∼TC(Y;p,Z/pν)
Now consider a smooth finite type scheme X X → X rarrX \rightarrowX→ Spec Λ Î› Lambda\LambdaΛ with special fiber i : Y X i : Y → X i:Y rarr Xi: Y \rightarrow Xi:Y→X and generic fiber j : V X j : V → X j:V rarr Xj: V \rightarrow Xj:V→X, as before.
Theorem 6.8 (Geisser-Hesselholt [51, THEorems A and B]). Suppose Λ Î› Lambda\LambdaΛ is henselian.
A. Suppose X Spec Λ X → Spec ⁡ Λ X rarr Spec LambdaX \rightarrow \operatorname{Spec} \LambdaX→Spec⁡Λ is smooth and proper. Then
trc : K q e t ( X , Z / p ν ) T C q ( X ; p , Z / p ν ) trc : K q e t X , Z / p ν → T C q X ; p , Z / p ν trc:K_(q)^(et)(X,Z//p^(nu))rarrTC_(q)(X;p,Z//p^(nu))\operatorname{trc}: K_{q}^{e t}\left(X, \mathbb{Z} / p^{\nu}\right) \rightarrow \mathrm{TC}_{q}\left(X ; p, \mathbb{Z} / p^{\nu}\right)trc:Kqet(X,Z/pν)→TCq(X;p,Z/pν)
is an isomorphism for all q Z q ∈ Z q inZq \in \mathbb{Z}q∈Z and v 1 v ≥ 1 v >= 1v \geq 1v≥1.
B. Suppose that X Spec Λ X → Spec ⁡ Λ X rarr Spec LambdaX \rightarrow \operatorname{Spec} \LambdaX→Spec⁡Λ is smooth and finite type. Then the map of prosheaves on Y e ́ t Y e ́ t Y_(ét)Y_{e ́ t}Yét,
i K q ( Z / p ν ) { i T C q m ( p , Z / p ν ) } m N i ∗ K q Z / p ν → i ∗ T C q m p , Z / p ν m ∈ N i^(**)K_(q)(Z//p^(nu))rarr{i^(**)TC_(q)^(m)(p,Z//p^(nu))}_(m inN)i^{*} \mathcal{K}_{q}\left(\mathbb{Z} / p^{\nu}\right) \rightarrow\left\{i^{*} \mathcal{T} \mathcal{C}_{q}^{m}\left(p, \mathbb{Z} / p^{\nu}\right)\right\}_{m \in \mathbb{N}}i∗Kq(Z/pν)→{i∗TCqm(p,Z/pν)}m∈N
is an isomorphism for all q Z q ∈ Z q inZq \in \mathbb{Z}q∈Z and all ν 1 ν ≥ 1 nu >= 1\nu \geq 1ν≥1.
Remark 6.9. To pass from the isomorphism of Theorem 6.7 to that of Theorem 6.8(B), Geisser-Hesselholt rely on the theorem of McCarthy [88], stating that the cyclotomic trace map from relative K K KKK-theory to relative TC,
trc : K q ( X / π n , X / π n r , Z / p ν ) T C q ( X / π n , X / π n r , Z / p ν ) trc : K q X / Ï€ n , X / Ï€ n − r , Z / p ν → T C q X / Ï€ n , X / Ï€ n − r , Z / p ν trc:K_(q)(X//pi^(n),X//pi^(n-r),Z//p^(nu))rarrTC_(q)(X//pi^(n),X//pi^(n-r),Z//p^(nu))\operatorname{trc}: K_{q}\left(X / \pi^{n}, X / \pi^{n-r}, \mathbb{Z} / p^{\nu}\right) \rightarrow \mathrm{TC}_{q}\left(X / \pi^{n}, X / \pi^{n-r}, \mathbb{Z} / p^{\nu}\right)trc:Kq(X/Ï€n,X/Ï€n−r,Z/pν)→TCq(X/Ï€n,X/Ï€n−r,Z/pν)
is an isomorphism for affine X X XXX. Thus, the K K KKK-theory and topological cyclic homology of non-reduced schemes play a central role in the proof of Theorem 6.8.

6.4. Integral p p ppp-adic Hodge theory and the motivic filtration

Bhatt-Morrow-Scholze [17,18] have constructed integral versions of p p ppp-adic Hodge theory. Here we discuss some aspects of the theory of [18] and its relation to p p ppp-adic étale motivic cohomology. This uses ( p p ppp-completed) topological Hochschild homology THH ( , Z p ) THH ⁡ − , Z p THH(-,Z_(p))\operatorname{THH}\left(-, \mathbb{Z}_{p}\right)THH⁡(−,Zp), topological negative cyclic homology TC ( , Z p ) TC − ⁡ − , Z p TC^(-)(-,Z_(p))\operatorname{TC}^{-}\left(-, \mathbb{Z}_{p}\right)TC−⁡(−,Zp), and topological periodic cyclic homology T P ( , Z p ) T P − , Z p TP(-,Z_(p))\mathrm{TP}\left(-, \mathbb{Z}_{p}\right)TP(−,Zp). For a nice, quick overview of these theories, we refer the reader to [ 18 , $ 1.2 , $ 2.3 ] [ 18 , $ 1.2 , $ 2.3 ] [18,$1.2,$2.3][18, \$ 1.2, \$ 2.3][18,$1.2,$2.3], and to [ 18 18 [18:}\left[18\right.[18, THEOREM 1.12] for their relation to T C ( , Z p ) T C − , Z p TC(-,Z_(p))\mathrm{TC}\left(-, \mathbb{Z}_{p}\right)TC(−,Zp).
Let C p C p C_(p)\mathbb{C}_{p}Cp be the completion of the algebraic closure of Q p Q p Q_(p)\mathbb{Q}_{p}Qp, with ring of integers O C p O C p O_(C_(p))\mathcal{O}_{\mathbb{C}_{p}}OCp. As in our review of the work of Fontaine-Messing, we have the F p F p F_(p)\mathbb{F}_{p}Fp-algebra O C p / p O C p / p O_(C_(p))//p\mathcal{O}_{\mathbb{C}_{p}} / pOCp/p, its perfection O C p b O C p b O_(C_(p))^(b)\mathcal{O}_{\mathbb{C}_{p}}^{b}OCpb and the ring of Witt vectors A inf ( O C p ) := W ( O C p b ) A inf  O C p := W O C p b A_("inf ")(O_(C_(p))):=W(O_(C_(p))^(b))A_{\text {inf }}\left(\mathcal{O}_{\mathbb{C}_{p}}\right):=W\left(\mathcal{O}_{\mathbb{C}_{p}}^{b}\right)Ainf (OCp):=W(OCpb). Hesselholt has connected this with negative cyclic homology T C T C − TC^(-)\mathrm{TC}^{-}TC−, constructing an isomorphism
π 0 T C ( O C p , Z p ) A i n f ( O C p ) Ï€ 0 T C − O C p , Z p ≅ A i n f O C p pi_(0)TC^(-)(O_(C_(p)),Z_(p))~=A_(inf)(O_(C_(p)))\pi_{0} \mathrm{TC}^{-}\left(\mathcal{O}_{\mathbb{C}_{p}}, \mathbb{Z}_{p}\right) \cong A_{\mathrm{inf}}\left(\mathcal{O}_{\mathbb{C}_{p}}\right)Ï€0TC−(OCp,Zp)≅Ainf(OCp)
This has been generalized by Bhatt-Morrow-Scholze in the setting of perfectoid rings (see [17, DEFINITION 3.5]). For a perfectoid ring R R RRR, we have Scholze's ring R b R b R^(b)R^{b}Rb, defined as for O C p b O C p b O_(C_(p))^(b)\mathcal{O}_{\mathbb{C}_{p}}^{b}OCpb by taking the perfection of R / p R / p R//pR / pR/p,
This gives the ring of Witt vectors A inf ( R ) := W ( R b ) A inf  ( R ) := W R b A_("inf ")(R):=W(R^(b))A_{\text {inf }}(R):=W\left(R^{b}\right)Ainf (R):=W(Rb) with Frobenius ϕ Ï• phi\phiÏ• induced by the Frobenius on R b R b R^(b)R^{b}Rb.
Theorem 6.10 (Bhatt-Morrow-Scholze [18, THEOREM 1.6]). Let R R RRR be a perfectoid ring. Then there is a canonical ϕ Ï• phi\phiÏ•-equivariant isomorphism π 0 T C ( R , Z p ) A inf ( R ) Ï€ 0 T C − R , Z p ≅ A inf  ( R ) pi_(0)TC^(-)(R,Z_(p))~=A_("inf ")(R)\pi_{0} \mathrm{TC}^{-}\left(R, \mathbb{Z}_{p}\right) \cong A_{\text {inf }}(R)Ï€0TC−(R,Zp)≅Ainf (R).
Fix a discretely valued extension K K KKK of Q p Q p Q_(p)\mathbb{Q}_{p}Qp, with ring of integers O K O K O_(K)\mathcal{O}_{K}OK having perfect residue field k k kkk. Let C C CCC be the completed algebraic closure of K K KKK, with ring of integers O C O C O_(C)\mathcal{O}_{C}OC. Let A inf := A inf ( O C ) A inf  := A inf  O C A_("inf "):=A_("inf ")(O_(C))A_{\text {inf }}:=A_{\text {inf }}\left(\mathcal{O}_{C}\right)Ainf :=Ainf (OC).
Let X X X\mathcal{X}X be a smooth formal scheme over O C O C O_(C)\mathcal{O}_{C}OC. In [17], Bhatt-Morrow-Scholze construct a presheaf of complexes of A i n f A i n f A_(inf)A_{\mathrm{inf}}Ainf-algebras on X Z a r , A Ω X X Z a r , A Ω X X_(Zar),AOmega_(X)\mathcal{X}_{\mathrm{Zar}}, A \Omega_{X}XZar,AΩX, whose Zariski hypercohomology specializes to crystalline cohomology, p p ppp-adic étale cohomology and de Rham cohomology via base-change with respect to suitable ring homomorphisms out of A inf A inf  A_("inf ")A_{\text {inf }}Ainf , replacing the ring homomorphism A inf ( O C p ) B crys A inf  O C p → B crys  A_("inf ")(O_(C_(p)))rarrB_("crys ")A_{\text {inf }}\left(\mathcal{O}_{\mathbb{C}_{p}}\right) \rightarrow B_{\text {crys }}Ainf (OCp)→Bcrys  used in the Fontaine-Messing theory. In [18], they refine and reinterpret this theory using T C T C − TC^(-)\mathrm{TC}^{-}TC−. They define the notion of a quasisyntomic ring and the associated quasi-syntomic site [18, DEFINITION 1.7]; this gives the presheaf π 0 T C ( ; Z p ) Ï€ 0 T C − − ; Z p pi_(0)TC^(-)(-;Z_(p))\pi_{0} \mathrm{TC}^{-}\left(-; \mathbb{Z}_{p}\right)Ï€0TC−(−;Zp) on the quasi-syntomic site q s y n A ~ q s y n A ~ qsyn_( tilde(A))\mathrm{qsyn}_{\tilde{A}}qsynA~ over a quasi-syntomic ring A ~ A ~ tilde(A)\tilde{A}A~ and the associated derived global sections functor R Γ syn ( A ~ , ) R Γ syn  ( A ~ , − ) RGamma_("syn ")( tilde(A),-)R \Gamma_{\text {syn }}(\tilde{A},-)RΓsyn (A~,−).
Theorem 6.11 ([18, THEOREM 1.8]). Let A ~ A ~ tilde(A)\tilde{A}A~ be an O C O C O_(C)\mathcal{O}_{C}OC-algebra that can be written as the p p ppp-adic completion of a smooth O C O C O_(C)\mathcal{O}_{C}OC-algebra. There is a functorial (in A ~ A ~ tilde(A)\tilde{A}A~ ) ϕ Ï• phi\phiÏ•-equivariant isomorphism of E A inf E ∞ − A inf  E_(oo)-A_("inf ")E_{\infty}-A_{\text {inf }}E∞−Ainf -algebras
A Ω A ~ R Γ s y n ( A ~ , π 0 T C ( ; Z p ) ) A Ω A ~ ≅ R Γ s y n A ~ , Ï€ 0 T C − − ; Z p AOmega_( tilde(A))~=RGamma_(syn)(( tilde(A)),pi_(0)TC^(-)(-;Z_(p)))A \Omega_{\tilde{A}} \cong R \Gamma_{\mathrm{syn}}\left(\tilde{A}, \pi_{0} \mathrm{TC}^{-}\left(-; \mathbb{Z}_{p}\right)\right)AΩA~≅RΓsyn(A~,Ï€0TC−(−;Zp))
The Postnikov tower τ T C ( ; Z p ) Ï„ ≥ ∗ T C − ; Z p tau_( >= **)TC(-;Z_(p))\tau_{\geq *} \mathrm{TC}\left(-; \mathbb{Z}_{p}\right)τ≥∗TC(−;Zp) for the presheaf of spectra T C ( ; Z p ) T C − ; Z p TC(-;Z_(p))\mathrm{TC}\left(-; \mathbb{Z}_{p}\right)TC(−;Zp) on A ~ qsyn A ~ qsyn  tilde(A)_("qsyn ")\tilde{A}_{\text {qsyn }}A~qsyn  induces the tower over TC ( A ~ ; Z p TC ⁡ A ~ ; Z p TC(( tilde(A));Z_(p):}\operatorname{TC}\left(\tilde{A} ; \mathbb{Z}_{p}\right.TC⁡(A~;Zp :
(6.5) Fil n + 1 T C ( A ~ ; Z p ) Fil n T C ( A ~ ; Z p ) TC ( A ~ ; Z p (6.5) ⋯ → Fil n + 1 ⁡ T C A ~ ; Z p → Fil n ⁡ T C A ~ ; Z p → ⋯ → TC ⁡ A ~ ; Z p {:(6.5)cdots rarrFil^(n+1)TC(( tilde(A));Z_(p))rarrFil^(n)TC(( tilde(A));Z_(p))rarr cdots rarr TC(( tilde(A));Z_(p):}:}\begin{equation*} \cdots \rightarrow \operatorname{Fil}^{n+1} \mathrm{TC}\left(\tilde{A} ; \mathbb{Z}_{p}\right) \rightarrow \operatorname{Fil}^{n} \mathrm{TC}\left(\tilde{A} ; \mathbb{Z}_{p}\right) \rightarrow \cdots \rightarrow \operatorname{TC}\left(\tilde{A} ; \mathbb{Z}_{p}\right. \tag{6.5} \end{equation*}(6.5)⋯→Filn+1⁡TC(A~;Zp)→Filn⁡TC(A~;Zp)→⋯→TC⁡(A~;Zp
by setting
Fil n T C ( A ~ ; Z p ) := R Γ syn ( A ~ , τ 2 n T C ( ; Z p ) ) Fil n ⁡ T C A ~ ; Z p := R Γ syn  A ~ , Ï„ ≥ 2 n T C − ; Z p Fil^(n)TC(( tilde(A));Z_(p)):=RGamma_("syn ")(( tilde(A)),tau_( >= 2n)TC(-;Z_(p)))\operatorname{Fil}^{n} \mathrm{TC}\left(\tilde{A} ; \mathbb{Z}_{p}\right):=R \Gamma_{\text {syn }}\left(\tilde{A}, \tau_{\geq 2 n} \mathrm{TC}\left(-; \mathbb{Z}_{p}\right)\right)Filn⁡TC(A~;Zp):=RΓsyn (A~,τ≥2nTC(−;Zp))
(see [ 18 , $ 1.4 ] ) [ 18 , $ 1.4 ] ) [18,$1.4])[18, \$ 1.4])[18,$1.4]). Define the sheaves Z p B M S ( n ) Z p B M S ( n ) Z_(p)^(BMS)(n)\mathbb{Z}_{p}^{\mathrm{BMS}}(n)ZpBMS(n) by sheafifying the presheaf
A ~ Z p B M S ( n ) ( A ~ ) := gr F i l n T C ( A ~ ; Z p ) [ 2 n ] A ~ ↦ Z p B M S ( n ) ( A ~ ) := gr F i l n ⁡ T C A ~ ; Z p [ − 2 n ] tilde(A)|->Z_(p)^(BMS)(n)( tilde(A)):=gr_(Fil)^(n)TC(( tilde(A));Z_(p))[-2n]\tilde{A} \mapsto \mathbb{Z}_{p}^{\mathrm{BMS}}(n)(\tilde{A}):=\operatorname{gr}_{\mathrm{Fil}}^{n} \mathrm{TC}\left(\tilde{A} ; \mathbb{Z}_{p}\right)[-2 n]A~↦ZpBMS(n)(A~):=grFiln⁡TC(A~;Zp)[−2n]
where gr F i l n T C ( A ~ ; Z p ) gr F i l n ⁡ T C A ~ ; Z p gr_(Fil)^(n)TC(( tilde(A));Z_(p))\operatorname{gr}_{\mathrm{Fil}}^{n} \mathrm{TC}\left(\tilde{A} ; \mathbb{Z}_{p}\right)grFiln⁡TC(A~;Zp) is the homotopy cofiber of Fil n + 1 T C ( A ~ ; Z p ) Fil n T C ( A ~ ; Z p ) Fil n + 1 ⁡ T C A ~ ; Z p → Fil n ⁡ T C A ~ ; Z p Fil^(n+1)TC(( tilde(A));Z_(p))rarrFil^(n)TC(( tilde(A));Z_(p))\operatorname{Fil}^{n+1} \mathrm{TC}\left(\tilde{A} ; \mathbb{Z}_{p}\right) \rightarrow \operatorname{Fil}^{n} \mathrm{TC}\left(\tilde{A} ; \mathbb{Z}_{p}\right)Filn+1⁡TC(A~;Zp)→Filn⁡TC(A~;Zp).
Theorem 6.12 ( [ 18 [ 18 [18[18[18, THEOREM 1.15]).
(1) Let k k kkk be a perfect field of characteristic p > 0 p > 0 p > 0p>0p>0, let A A AAA be a smooth k k kkk-algebra and let X = Spec A X = Spec ⁡ A X=Spec AX=\operatorname{Spec} AX=Spec⁡A. Then there is an isomorphism in the derived category of sheaves on the pro-étale site of X X XXX,
Z p B M S ( r ) X W Ω X , log r [ r ] Z p B M S ( r ) X ≅ W Ω X , log r [ − r ] Z_(p)^(BMS)(r)_(X)~=WOmega_(X,log)^(r)[-r]\mathbb{Z}_{p}^{\mathrm{BMS}}(r)_{X} \cong W \Omega_{X, \log }^{r}[-r]ZpBMS(r)X≅WΩX,logr[−r]
(2) Let C C CCC be an algebraically closed complete extension of Q p Q p Q_(p)\mathbb{Q}_{p}Qp, let A A AAA be the completion of a smooth O C O C O_(C)\mathcal{O}_{C}OC-algebra, and let X = Spf A X = Spf ⁡ A X=Spf A\mathcal{X}=\operatorname{Spf} AX=Spf⁡A. Then there is an isomorphism in the derived category of sheaves on the pro-étale site of X X X\mathcal{X}X,
Z p B M S ( r ) X τ r R ψ Z p ( r ) X , e ́ t Z p B M S ( r ) X ≅ Ï„ ≤ r R ψ Z p ( r ) X , e ́ t Z_(p)^(BMS)(r)X~=tau_( <= r)R psiZ_(p)(r)_(X,ét)\mathbb{Z}_{p}^{\mathrm{BMS}}(r) X \cong \tau_{\leq r} R \psi \mathbb{Z}_{p}(r)_{X, e ́ t}ZpBMS(r)X≅τ≤rRψZp(r)X,ét
Here Z p ( r ) X , ét Z p ( r ) X ,  ét  Z_(p)(r)_(X," ét ")\mathbb{Z}_{p}(r)_{X, \text { ét }}Zp(r)X, ét  denotes the pro-étale sheaf